Complete Directed Cuts and Equal-Value Flow Optimality
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batch-20261015-advance-20261115-r05-max-flow-min-cut-theorem-foundations · 2026-10-15 · 100 questions
This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.
Field order uses a source-bound presentation supplement; canonical JSONL and mathematical answers are unchanged.
Field-order and source-binding data
1. Every Source-Side Vertex Subset Defines One Directed Cut Audit
Use complete directed vertices=[s, a0, a1, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t], [a0, s]], source=s, sink=t, aligned nonnegative exact capacities=[2, 2, 3, 3, 7]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (vertices, source, sink, cut_rows, cut_count, required_cut_count, certificate).
View answer and explanation
([a0, a1, s, t], s, t, [[[s], [[[s, a0], 2], [[s, a1], 3]], 5], [[a0, s], [[[a0, t], 2], [[s, a1], 3]], 5], [[a1, s], [[[a1, t], 3], [[s, a0], 2]], 5], [[a0, a1, s], [[[a0, t], 2], [[a1, t], 3]], 5]], 4, 4, every_subset_containing_s_and_excluding_t_is_enumerated_and_only_forward_edges_are_counted)
Enumerate every subset of the other vertices, with source included and sink excluded. For each cut list every outgoing edge and its capacity, then sum exactly. Retain actual and required counts; incoming edges are not charged to cut capacity. The exact result, retaining the requested order, is ([a0, a1, s, t], s, t, [[[s], [[[s, a0], 2], [[s, a1], 3]], 5], [[a0, s], [[[a0, t], 2], [[s, a1], 3]], 5], [[a1, s], [[[a1, t], 3], [[s, a0], 2]], 5], [[a0, a1, s], [[[a0, t], 2], [[a1, t], 3]], 5]], 4, 4, every_subset_containing_s_and_excluding_t_is_enumerated_and_only_forward_edges_are_counted).
Reviewed step-by-step derivation
Max-flow min-cut theorem · qf-mod46-mfmc-complete-cut-enumeration · candidate
2. Every Source-Side Vertex Subset Defines One Directed Cut Audit
Use complete directed vertices=[s, a0, a1, a2, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t], [s, a2], [a2, t]], source=s, sink=t, aligned nonnegative exact capacities=[3, 3, 4, 4, 5, 5]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (vertices, source, sink, cut_rows, cut_count, required_cut_count, certificate).
View answer and explanation
([a0, a1, a2, s, t], s, t, [[[s], [[[s, a0], 3], [[s, a1], 4], [[s, a2], 5]], 12], [[a0, s], [[[a0, t], 3], [[s, a1], 4], [[s, a2], 5]], 12], [[a1, s], [[[a1, t], 4], [[s, a0], 3], [[s, a2], 5]], 12], [[a0, a1, s], [[[a0, t], 3], [[a1, t], 4], [[s, a2], 5]], 12], [[a2, s], [[[a2, t], 5], [[s, a0], 3], [[s, a1], 4]], 12], [[a0, a2, s], [[[a0, t], 3], [[a2, t], 5], [[s, a1], 4]], 12], [[a1, a2, s], [[[a1, t], 4], [[a2, t], 5], [[s, a0], 3]], 12], [[a0, a1, a2, s], [[[a0, t], 3], [[a1, t], 4], [[a2, t], 5]], 12]], 8, 8, every_subset_containing_s_and_excluding_t_is_enumerated_and_only_forward_edges_are_counted)
Enumerate every subset of the other vertices, with source included and sink excluded. For each cut list every outgoing edge and its capacity, then sum exactly. Retain actual and required counts; incoming edges are not charged to cut capacity. The exact result, retaining the requested order, is ([a0, a1, a2, s, t], s, t, [[[s], [[[s, a0], 3], [[s, a1], 4], [[s, a2], 5]], 12], [[a0, s], [[[a0, t], 3], [[s, a1], 4], [[s, a2], 5]], 12], [[a1, s], [[[a1, t], 4], [[s, a0], 3], [[s, a2], 5]], 12], [[a0, a1, s], [[[a0, t], 3], [[a1, t], 4], [[s, a2], 5]], 12], [[a2, s], [[[a2, t], 5], [[s, a0], 3], [[s, a1], 4]], 12], [[a0, a2, s], [[[a0, t], 3], [[a2, t], 5], [[s, a1], 4]], 12], [[a1, a2, s], [[[a1, t], 4], [[a2, t], 5], [[s, a0], 3]], 12], [[a0, a1, a2, s], [[[a0, t], 3], [[a1, t], 4], [[a2, t], 5]], 12]], 8, 8, every_subset_containing_s_and_excluding_t_is_enumerated_and_only_forward_edges_are_counted).
Reviewed step-by-step derivation
Max-flow min-cut theorem · qf-mod46-mfmc-complete-cut-enumeration · candidate
3. Every Source-Side Vertex Subset Defines One Directed Cut Audit
Use complete directed vertices=[s, a0, a1, a2, a3, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t], [s, a2], [a2, t], [s, a3], [a3, t]], source=s, sink=t, aligned nonnegative exact capacities=[4, 4, 5, 5, 6, 6, 7, 7]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (vertices, source, sink, cut_rows, cut_count, required_cut_count, certificate).
View answer and explanation
([a0, a1, a2, a3, s, t], s, t, [[[s], [[[s, a0], 4], [[s, a1], 5], [[s, a2], 6], [[s, a3], 7]], 22], [[a0, s], [[[a0, t], 4], [[s, a1], 5], [[s, a2], 6], [[s, a3], 7]], 22], [[a1, s], [[[a1, t], 5], [[s, a0], 4], [[s, a2], 6], [[s, a3], 7]], 22], [[a0, a1, s], [[[a0, t], 4], [[a1, t], 5], [[s, a2], 6], [[s, a3], 7]], 22], [[a2, s], [[[a2, t], 6], [[s, a0], 4], [[s, a1], 5], [[s, a3], 7]], 22], [[a0, a2, s], [[[a0, t], 4], [[a2, t], 6], [[s, a1], 5], [[s, a3], 7]], 22], [[a1, a2, s], [[[a1, t], 5], [[a2, t], 6], [[s, a0], 4], [[s, a3], 7]], 22], [[a0, a1, a2, s], [[[a0, t], 4], [[a1, t], 5], [[a2, t], 6], [[s, a3], 7]], 22], [[a3, s], [[[a3, t], 7], [[s, a0], 4], [[s, a1], 5], [[s, a2], 6]], 22], [[a0, a3, s], [[[a0, t], 4], [[a3, t], 7], [[s, a1], 5], [[s, a2], 6]], 22], [[a1, a3, s], [[[a1, t], 5], [[a3, t], 7], [[s, a0], 4], [[s, a2], 6]], 22], [[a0, a1, a3, s], [[[a0, t], 4], [[a1, t], 5], [[a3, t], 7], [[s, a2], 6]], 22], [[a2, a3, s], [[[a2, t], 6], [[a3, t], 7], [[s, a0], 4], [[s, a1], 5]], 22], [[a0, a2, a3, s], [[[a0, t], 4], [[a2, t], 6], [[a3, t], 7], [[s, a1], 5]], 22], [[a1, a2, a3, s], [[[a1, t], 5], [[a2, t], 6], [[a3, t], 7], [[s, a0], 4]], 22], [[a0, a1, a2, a3, s], [[[a0, t], 4], [[a1, t], 5], [[a2, t], 6], [[a3, t], 7]], 22]], 16, 16, every_subset_containing_s_and_excluding_t_is_enumerated_and_only_forward_edges_are_counted)
Enumerate every subset of the other vertices, with source included and sink excluded. For each cut list every outgoing edge and its capacity, then sum exactly. Retain actual and required counts; incoming edges are not charged to cut capacity. The exact result, retaining the requested order, is ([a0, a1, a2, a3, s, t], s, t, [[[s], [[[s, a0], 4], [[s, a1], 5], [[s, a2], 6], [[s, a3], 7]], 22], [[a0, s], [[[a0, t], 4], [[s, a1], 5], [[s, a2], 6], [[s, a3], 7]], 22], [[a1, s], [[[a1, t], 5], [[s, a0], 4], [[s, a2], 6], [[s, a3], 7]], 22], [[a0, a1, s], [[[a0, t], 4], [[a1, t], 5], [[s, a2], 6], [[s, a3], 7]], 22], [[a2, s], [[[a2, t], 6], [[s, a0], 4], [[s, a1], 5], [[s, a3], 7]], 22], [[a0, a2, s], [[[a0, t], 4], [[a2, t], 6], [[s, a1], 5], [[s, a3], 7]], 22], [[a1, a2, s], [[[a1, t], 5], [[a2, t], 6], [[s, a0], 4], [[s, a3], 7]], 22], [[a0, a1, a2, s], [[[a0, t], 4], [[a1, t], 5], [[a2, t], 6], [[s, a3], 7]], 22], [[a3, s], [[[a3, t], 7], [[s, a0], 4], [[s, a1], 5], [[s, a2], 6]], 22], [[a0, a3, s], [[[a0, t], 4], [[a3, t], 7], [[s, a1], 5], [[s, a2], 6]], 22], [[a1, a3, s], [[[a1, t], 5], [[a3, t], 7], [[s, a0], 4], [[s, a2], 6]], 22], [[a0, a1, a3, s], [[[a0, t], 4], [[a1, t], 5], [[a3, t], 7], [[s, a2], 6]], 22], [[a2, a3, s], [[[a2, t], 6], [[a3, t], 7], [[s, a0], 4], [[s, a1], 5]], 22], [[a0, a2, a3, s], [[[a0, t], 4], [[a2, t], 6], [[a3, t], 7], [[s, a1], 5]], 22], [[a1, a2, a3, s], [[[a1, t], 5], [[a2, t], 6], [[a3, t], 7], [[s, a0], 4]], 22], [[a0, a1, a2, a3, s], [[[a0, t], 4], [[a1, t], 5], [[a2, t], 6], [[a3, t], 7]], 22]], 16, 16, every_subset_containing_s_and_excluding_t_is_enumerated_and_only_forward_edges_are_counted).
Reviewed step-by-step derivation
Max-flow min-cut theorem · qf-mod46-mfmc-complete-cut-enumeration · candidate
4. Every Source-Side Vertex Subset Defines One Directed Cut Audit
Use complete directed vertices=[s, a0, a1, a2, a3, a4, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t], [s, a2], [a2, t], [s, a3], [a3, t], [s, a4], [a4, t], [a0, s]], source=s, sink=t, aligned nonnegative exact capacities=[5, 5, 6, 6, 7, 7, 8, 8, 9, 9, 10]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (vertices, source, sink, cut_rows, cut_count, required_cut_count, certificate).
View answer and explanation
([a0, a1, a2, a3, a4, s, t], s, t, [[[s], [[[s, a0], 5], [[s, a1], 6], [[s, a2], 7], [[s, a3], 8], [[s, a4], 9]], 35], [[a0, s], [[[a0, t], 5], [[s, a1], 6], [[s, a2], 7], [[s, a3], 8], [[s, a4], 9]], 35], [[a1, s], [[[a1, t], 6], [[s, a0], 5], [[s, a2], 7], [[s, a3], 8], [[s, a4], 9]], 35], [[a0, a1, s], [[[a0, t], 5], [[a1, t], 6], [[s, a2], 7], [[s, a3], 8], [[s, a4], 9]], 35], [[a2, s], [[[a2, t], 7], [[s, a0], 5], [[s, a1], 6], [[s, a3], 8], [[s, a4], 9]], 35], [[a0, a2, s], [[[a0, t], 5], [[a2, t], 7], [[s, a1], 6], [[s, a3], 8], [[s, a4], 9]], 35], [[a1, a2, s], [[[a1, t], 6], [[a2, t], 7], [[s, a0], 5], [[s, a3], 8], [[s, a4], 9]], 35], [[a0, a1, a2, s], [[[a0, t], 5], [[a1, t], 6], [[a2, t], 7], [[s, a3], 8], [[s, a4], 9]], 35], [[a3, s], [[[a3, t], 8], [[s, a0], 5], [[s, a1], 6], [[s, a2], 7], [[s, a4], 9]], 35], [[a0, a3, s], [[[a0, t], 5], [[a3, t], 8], [[s, a1], 6], [[s, a2], 7], [[s, a4], 9]], 35], [[a1, a3, s], [[[a1, t], 6], [[a3, t], 8], [[s, a0], 5], [[s, a2], 7], [[s, a4], 9]], 35], [[a0, a1, a3, s], [[[a0, t], 5], [[a1, t], 6], [[a3, t], 8], [[s, a2], 7], [[s, a4], 9]], 35], [[a2, a3, s], [[[a2, t], 7], [[a3, t], 8], [[s, a0], 5], [[s, a1], 6], [[s, a4], 9]], 35], [[a0, a2, a3, s], [[[a0, t], 5], [[a2, t], 7], [[a3, t], 8], [[s, a1], 6], [[s, a4], 9]], 35], [[a1, a2, a3, s], [[[a1, t], 6], [[a2, t], 7], [[a3, t], 8], [[s, a0], 5], [[s, a4], 9]], 35], [[a0, a1, a2, a3, s], [[[a0, t], 5], [[a1, t], 6], [[a2, t], 7], [[a3, t], 8], [[s, a4], 9]], 35], [[a4, s], [[[a4, t], 9], [[s, a0], 5], [[s, a1], 6], [[s, a2], 7], [[s, a3], 8]], 35], [[a0, a4, s], [[[a0, t], 5], [[a4, t], 9], [[s, a1], 6], [[s, a2], 7], [[s, a3], 8]], 35], [[a1, a4, s], [[[a1, t], 6], [[a4, t], 9], [[s, a0], 5], [[s, a2], 7], [[s, a3], 8]], 35], [[a0, a1, a4, s], [[[a0, t], 5], [[a1, t], 6], [[a4, t], 9], [[s, a2], 7], [[s, a3], 8]], 35], [[a2, a4, s], [[[a2, t], 7], [[a4, t], 9], [[s, a0], 5], [[s, a1], 6], [[s, a3], 8]], 35], [[a0, a2, a4, s], [[[a0, t], 5], [[a2, t], 7], [[a4, t], 9], [[s, a1], 6], [[s, a3], 8]], 35], [[a1, a2, a4, s], [[[a1, t], 6], [[a2, t], 7], [[a4, t], 9], [[s, a0], 5], [[s, a3], 8]], 35], [[a0, a1, a2, a4, s], [[[a0, t], 5], [[a1, t], 6], [[a2, t], 7], [[a4, t], 9], [[s, a3], 8]], 35], [[a3, a4, s], [[[a3, t], 8], [[a4, t], 9], [[s, a0], 5], [[s, a1], 6], [[s, a2], 7]], 35], [[a0, a3, a4, s], [[[a0, t], 5], [[a3, t], 8], [[a4, t], 9], [[s, a1], 6], [[s, a2], 7]], 35], [[a1, a3, a4, s], [[[a1, t], 6], [[a3, t], 8], [[a4, t], 9], [[s, a0], 5], [[s, a2], 7]], 35], [[a0, a1, a3, a4, s], [[[a0, t], 5], [[a1, t], 6], [[a3, t], 8], [[a4, t], 9], [[s, a2], 7]], 35], [[a2, a3, a4, s], [[[a2, t], 7], [[a3, t], 8], [[a4, t], 9], [[s, a0], 5], [[s, a1], 6]], 35], [[a0, a2, a3, a4, s], [[[a0, t], 5], [[a2, t], 7], [[a3, t], 8], [[a4, t], 9], [[s, a1], 6]], 35], [[a1, a2, a3, a4, s], [[[a1, t], 6], [[a2, t], 7], [[a3, t], 8], [[a4, t], 9], [[s, a0], 5]], 35], [[a0, a1, a2, a3, a4, s], [[[a0, t], 5], [[a1, t], 6], [[a2, t], 7], [[a3, t], 8], [[a4, t], 9]], 35]], 32, 32, every_subset_containing_s_and_excluding_t_is_enumerated_and_only_forward_edges_are_counted)
Enumerate every subset of the other vertices, with source included and sink excluded. For each cut list every outgoing edge and its capacity, then sum exactly. Retain actual and required counts; incoming edges are not charged to cut capacity. The exact result, retaining the requested order, is ([a0, a1, a2, a3, a4, s, t], s, t, [[[s], [[[s, a0], 5], [[s, a1], 6], [[s, a2], 7], [[s, a3], 8], [[s, a4], 9]], 35], [[a0, s], [[[a0, t], 5], [[s, a1], 6], [[s, a2], 7], [[s, a3], 8], [[s, a4], 9]], 35], [[a1, s], [[[a1, t], 6], [[s, a0], 5], [[s, a2], 7], [[s, a3], 8], [[s, a4], 9]], 35], [[a0, a1, s], [[[a0, t], 5], [[a1, t], 6], [[s, a2], 7], [[s, a3], 8], [[s, a4], 9]], 35], [[a2, s], [[[a2, t], 7], [[s, a0], 5], [[s, a1], 6], [[s, a3], 8], [[s, a4], 9]], 35], [[a0, a2, s], [[[a0, t], 5], [[a2, t], 7], [[s, a1], 6], [[s, a3], 8], [[s, a4], 9]], 35], [[a1, a2, s], [[[a1, t], 6], [[a2, t], 7], [[s, a0], 5], [[s, a3], 8], [[s, a4], 9]], 35], [[a0, a1, a2, s], [[[a0, t], 5], [[a1, t], 6], [[a2, t], 7], [[s, a3], 8], [[s, a4], 9]], 35], [[a3, s], [[[a3, t], 8], [[s, a0], 5], [[s, a1], 6], [[s, a2], 7], [[s, a4], 9]], 35], [[a0, a3, s], [[[a0, t], 5], [[a3, t], 8], [[s, a1], 6], [[s, a2], 7], [[s, a4], 9]], 35], [[a1, a3, s], [[[a1, t], 6], [[a3, t], 8], [[s, a0], 5], [[s, a2], 7], [[s, a4], 9]], 35], [[a0, a1, a3, s], [[[a0, t], 5], [[a1, t], 6], [[a3, t], 8], [[s, a2], 7], [[s, a4], 9]], 35], [[a2, a3, s], [[[a2, t], 7], [[a3, t], 8], [[s, a0], 5], [[s, a1], 6], [[s, a4], 9]], 35], [[a0, a2, a3, s], [[[a0, t], 5], [[a2, t], 7], [[a3, t], 8], [[s, a1], 6], [[s, a4], 9]], 35], [[a1, a2, a3, s], [[[a1, t], 6], [[a2, t], 7], [[a3, t], 8], [[s, a0], 5], [[s, a4], 9]], 35], [[a0, a1, a2, a3, s], [[[a0, t], 5], [[a1, t], 6], [[a2, t], 7], [[a3, t], 8], [[s, a4], 9]], 35], [[a4, s], [[[a4, t], 9], [[s, a0], 5], [[s, a1], 6], [[s, a2], 7], [[s, a3], 8]], 35], [[a0, a4, s], [[[a0, t], 5], [[a4, t], 9], [[s, a1], 6], [[s, a2], 7], [[s, a3], 8]], 35], [[a1, a4, s], [[[a1, t], 6], [[a4, t], 9], [[s, a0], 5], [[s, a2], 7], [[s, a3], 8]], 35], [[a0, a1, a4, s], [[[a0, t], 5], [[a1, t], 6], [[a4, t], 9], [[s, a2], 7], [[s, a3], 8]], 35], [[a2, a4, s], [[[a2, t], 7], [[a4, t], 9], [[s, a0], 5], [[s, a1], 6], [[s, a3], 8]], 35], [[a0, a2, a4, s], [[[a0, t], 5], [[a2, t], 7], [[a4, t], 9], [[s, a1], 6], [[s, a3], 8]], 35], [[a1, a2, a4, s], [[[a1, t], 6], [[a2, t], 7], [[a4, t], 9], [[s, a0], 5], [[s, a3], 8]], 35], [[a0, a1, a2, a4, s], [[[a0, t], 5], [[a1, t], 6], [[a2, t], 7], [[a4, t], 9], [[s, a3], 8]], 35], [[a3, a4, s], [[[a3, t], 8], [[a4, t], 9], [[s, a0], 5], [[s, a1], 6], [[s, a2], 7]], 35], [[a0, a3, a4, s], [[[a0, t], 5], [[a3, t], 8], [[a4, t], 9], [[s, a1], 6], [[s, a2], 7]], 35], [[a1, a3, a4, s], [[[a1, t], 6], [[a3, t], 8], [[a4, t], 9], [[s, a0], 5], [[s, a2], 7]], 35], [[a0, a1, a3, a4, s], [[[a0, t], 5], [[a1, t], 6], [[a3, t], 8], [[a4, t], 9], [[s, a2], 7]], 35], [[a2, a3, a4, s], [[[a2, t], 7], [[a3, t], 8], [[a4, t], 9], [[s, a0], 5], [[s, a1], 6]], 35], [[a0, a2, a3, a4, s], [[[a0, t], 5], [[a2, t], 7], [[a3, t], 8], [[a4, t], 9], [[s, a1], 6]], 35], [[a1, a2, a3, a4, s], [[[a1, t], 6], [[a2, t], 7], [[a3, t], 8], [[a4, t], 9], [[s, a0], 5]], 35], [[a0, a1, a2, a3, a4, s], [[[a0, t], 5], [[a1, t], 6], [[a2, t], 7], [[a3, t], 8], [[a4, t], 9]], 35]], 32, 32, every_subset_containing_s_and_excluding_t_is_enumerated_and_only_forward_edges_are_counted).
Reviewed step-by-step derivation
Max-flow min-cut theorem · qf-mod46-mfmc-complete-cut-enumeration · candidate
5. Every Source-Side Vertex Subset Defines One Directed Cut Audit
Use complete directed vertices=[s, a0, a1, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t]], source=s, sink=t, aligned nonnegative exact capacities=[6, 6, 7, 7]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (vertices, source, sink, cut_rows, cut_count, required_cut_count, certificate).
View answer and explanation
([a0, a1, s, t], s, t, [[[s], [[[s, a0], 6], [[s, a1], 7]], 13], [[a0, s], [[[a0, t], 6], [[s, a1], 7]], 13], [[a1, s], [[[a1, t], 7], [[s, a0], 6]], 13], [[a0, a1, s], [[[a0, t], 6], [[a1, t], 7]], 13]], 4, 4, every_subset_containing_s_and_excluding_t_is_enumerated_and_only_forward_edges_are_counted)
Enumerate every subset of the other vertices, with source included and sink excluded. For each cut list every outgoing edge and its capacity, then sum exactly. Retain actual and required counts; incoming edges are not charged to cut capacity. The exact result, retaining the requested order, is ([a0, a1, s, t], s, t, [[[s], [[[s, a0], 6], [[s, a1], 7]], 13], [[a0, s], [[[a0, t], 6], [[s, a1], 7]], 13], [[a1, s], [[[a1, t], 7], [[s, a0], 6]], 13], [[a0, a1, s], [[[a0, t], 6], [[a1, t], 7]], 13]], 4, 4, every_subset_containing_s_and_excluding_t_is_enumerated_and_only_forward_edges_are_counted).
Reviewed step-by-step derivation
Max-flow min-cut theorem · qf-mod46-mfmc-complete-cut-enumeration · candidate
6. Every Source-Side Vertex Subset Defines One Directed Cut Audit
Use complete directed vertices=[s, a0, a1, a2, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t], [s, a2], [a2, t]], source=s, sink=t, aligned nonnegative exact capacities=[7, 7, 8, 8, 9, 9]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (vertices, source, sink, cut_rows, cut_count, required_cut_count, certificate).
View answer and explanation
([a0, a1, a2, s, t], s, t, [[[s], [[[s, a0], 7], [[s, a1], 8], [[s, a2], 9]], 24], [[a0, s], [[[a0, t], 7], [[s, a1], 8], [[s, a2], 9]], 24], [[a1, s], [[[a1, t], 8], [[s, a0], 7], [[s, a2], 9]], 24], [[a0, a1, s], [[[a0, t], 7], [[a1, t], 8], [[s, a2], 9]], 24], [[a2, s], [[[a2, t], 9], [[s, a0], 7], [[s, a1], 8]], 24], [[a0, a2, s], [[[a0, t], 7], [[a2, t], 9], [[s, a1], 8]], 24], [[a1, a2, s], [[[a1, t], 8], [[a2, t], 9], [[s, a0], 7]], 24], [[a0, a1, a2, s], [[[a0, t], 7], [[a1, t], 8], [[a2, t], 9]], 24]], 8, 8, every_subset_containing_s_and_excluding_t_is_enumerated_and_only_forward_edges_are_counted)
Enumerate every subset of the other vertices, with source included and sink excluded. For each cut list every outgoing edge and its capacity, then sum exactly. Retain actual and required counts; incoming edges are not charged to cut capacity. The exact result, retaining the requested order, is ([a0, a1, a2, s, t], s, t, [[[s], [[[s, a0], 7], [[s, a1], 8], [[s, a2], 9]], 24], [[a0, s], [[[a0, t], 7], [[s, a1], 8], [[s, a2], 9]], 24], [[a1, s], [[[a1, t], 8], [[s, a0], 7], [[s, a2], 9]], 24], [[a0, a1, s], [[[a0, t], 7], [[a1, t], 8], [[s, a2], 9]], 24], [[a2, s], [[[a2, t], 9], [[s, a0], 7], [[s, a1], 8]], 24], [[a0, a2, s], [[[a0, t], 7], [[a2, t], 9], [[s, a1], 8]], 24], [[a1, a2, s], [[[a1, t], 8], [[a2, t], 9], [[s, a0], 7]], 24], [[a0, a1, a2, s], [[[a0, t], 7], [[a1, t], 8], [[a2, t], 9]], 24]], 8, 8, every_subset_containing_s_and_excluding_t_is_enumerated_and_only_forward_edges_are_counted).
Reviewed step-by-step derivation
Max-flow min-cut theorem · qf-mod46-mfmc-complete-cut-enumeration · candidate
7. Every Source-Side Vertex Subset Defines One Directed Cut Audit
Use complete directed vertices=[s, a0, a1, a2, a3, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t], [s, a2], [a2, t], [s, a3], [a3, t], [a0, s]], source=s, sink=t, aligned nonnegative exact capacities=[8, 8, 9, 9, 10, 10, 11, 11, 13]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (vertices, source, sink, cut_rows, cut_count, required_cut_count, certificate).
View answer and explanation
([a0, a1, a2, a3, s, t], s, t, [[[s], [[[s, a0], 8], [[s, a1], 9], [[s, a2], 10], [[s, a3], 11]], 38], [[a0, s], [[[a0, t], 8], [[s, a1], 9], [[s, a2], 10], [[s, a3], 11]], 38], [[a1, s], [[[a1, t], 9], [[s, a0], 8], [[s, a2], 10], [[s, a3], 11]], 38], [[a0, a1, s], [[[a0, t], 8], [[a1, t], 9], [[s, a2], 10], [[s, a3], 11]], 38], [[a2, s], [[[a2, t], 10], [[s, a0], 8], [[s, a1], 9], [[s, a3], 11]], 38], [[a0, a2, s], [[[a0, t], 8], [[a2, t], 10], [[s, a1], 9], [[s, a3], 11]], 38], [[a1, a2, s], [[[a1, t], 9], [[a2, t], 10], [[s, a0], 8], [[s, a3], 11]], 38], [[a0, a1, a2, s], [[[a0, t], 8], [[a1, t], 9], [[a2, t], 10], [[s, a3], 11]], 38], [[a3, s], [[[a3, t], 11], [[s, a0], 8], [[s, a1], 9], [[s, a2], 10]], 38], [[a0, a3, s], [[[a0, t], 8], [[a3, t], 11], [[s, a1], 9], [[s, a2], 10]], 38], [[a1, a3, s], [[[a1, t], 9], [[a3, t], 11], [[s, a0], 8], [[s, a2], 10]], 38], [[a0, a1, a3, s], [[[a0, t], 8], [[a1, t], 9], [[a3, t], 11], [[s, a2], 10]], 38], [[a2, a3, s], [[[a2, t], 10], [[a3, t], 11], [[s, a0], 8], [[s, a1], 9]], 38], [[a0, a2, a3, s], [[[a0, t], 8], [[a2, t], 10], [[a3, t], 11], [[s, a1], 9]], 38], [[a1, a2, a3, s], [[[a1, t], 9], [[a2, t], 10], [[a3, t], 11], [[s, a0], 8]], 38], [[a0, a1, a2, a3, s], [[[a0, t], 8], [[a1, t], 9], [[a2, t], 10], [[a3, t], 11]], 38]], 16, 16, every_subset_containing_s_and_excluding_t_is_enumerated_and_only_forward_edges_are_counted)
Enumerate every subset of the other vertices, with source included and sink excluded. For each cut list every outgoing edge and its capacity, then sum exactly. Retain actual and required counts; incoming edges are not charged to cut capacity. The exact result, retaining the requested order, is ([a0, a1, a2, a3, s, t], s, t, [[[s], [[[s, a0], 8], [[s, a1], 9], [[s, a2], 10], [[s, a3], 11]], 38], [[a0, s], [[[a0, t], 8], [[s, a1], 9], [[s, a2], 10], [[s, a3], 11]], 38], [[a1, s], [[[a1, t], 9], [[s, a0], 8], [[s, a2], 10], [[s, a3], 11]], 38], [[a0, a1, s], [[[a0, t], 8], [[a1, t], 9], [[s, a2], 10], [[s, a3], 11]], 38], [[a2, s], [[[a2, t], 10], [[s, a0], 8], [[s, a1], 9], [[s, a3], 11]], 38], [[a0, a2, s], [[[a0, t], 8], [[a2, t], 10], [[s, a1], 9], [[s, a3], 11]], 38], [[a1, a2, s], [[[a1, t], 9], [[a2, t], 10], [[s, a0], 8], [[s, a3], 11]], 38], [[a0, a1, a2, s], [[[a0, t], 8], [[a1, t], 9], [[a2, t], 10], [[s, a3], 11]], 38], [[a3, s], [[[a3, t], 11], [[s, a0], 8], [[s, a1], 9], [[s, a2], 10]], 38], [[a0, a3, s], [[[a0, t], 8], [[a3, t], 11], [[s, a1], 9], [[s, a2], 10]], 38], [[a1, a3, s], [[[a1, t], 9], [[a3, t], 11], [[s, a0], 8], [[s, a2], 10]], 38], [[a0, a1, a3, s], [[[a0, t], 8], [[a1, t], 9], [[a3, t], 11], [[s, a2], 10]], 38], [[a2, a3, s], [[[a2, t], 10], [[a3, t], 11], [[s, a0], 8], [[s, a1], 9]], 38], [[a0, a2, a3, s], [[[a0, t], 8], [[a2, t], 10], [[a3, t], 11], [[s, a1], 9]], 38], [[a1, a2, a3, s], [[[a1, t], 9], [[a2, t], 10], [[a3, t], 11], [[s, a0], 8]], 38], [[a0, a1, a2, a3, s], [[[a0, t], 8], [[a1, t], 9], [[a2, t], 10], [[a3, t], 11]], 38]], 16, 16, every_subset_containing_s_and_excluding_t_is_enumerated_and_only_forward_edges_are_counted).
Reviewed step-by-step derivation
Max-flow min-cut theorem · qf-mod46-mfmc-complete-cut-enumeration · candidate
8. Every Source-Side Vertex Subset Defines One Directed Cut Audit
Use complete directed vertices=[s, a0, a1, a2, a3, a4, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t], [s, a2], [a2, t], [s, a3], [a3, t], [s, a4], [a4, t]], source=s, sink=t, aligned nonnegative exact capacities=[9, 9, 10, 10, 11, 11, 12, 12, 13, 13]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (vertices, source, sink, cut_rows, cut_count, required_cut_count, certificate).
View answer and explanation
([a0, a1, a2, a3, a4, s, t], s, t, [[[s], [[[s, a0], 9], [[s, a1], 10], [[s, a2], 11], [[s, a3], 12], [[s, a4], 13]], 55], [[a0, s], [[[a0, t], 9], [[s, a1], 10], [[s, a2], 11], [[s, a3], 12], [[s, a4], 13]], 55], [[a1, s], [[[a1, t], 10], [[s, a0], 9], [[s, a2], 11], [[s, a3], 12], [[s, a4], 13]], 55], [[a0, a1, s], [[[a0, t], 9], [[a1, t], 10], [[s, a2], 11], [[s, a3], 12], [[s, a4], 13]], 55], [[a2, s], [[[a2, t], 11], [[s, a0], 9], [[s, a1], 10], [[s, a3], 12], [[s, a4], 13]], 55], [[a0, a2, s], [[[a0, t], 9], [[a2, t], 11], [[s, a1], 10], [[s, a3], 12], [[s, a4], 13]], 55], [[a1, a2, s], [[[a1, t], 10], [[a2, t], 11], [[s, a0], 9], [[s, a3], 12], [[s, a4], 13]], 55], [[a0, a1, a2, s], [[[a0, t], 9], [[a1, t], 10], [[a2, t], 11], [[s, a3], 12], [[s, a4], 13]], 55], [[a3, s], [[[a3, t], 12], [[s, a0], 9], [[s, a1], 10], [[s, a2], 11], [[s, a4], 13]], 55], [[a0, a3, s], [[[a0, t], 9], [[a3, t], 12], [[s, a1], 10], [[s, a2], 11], [[s, a4], 13]], 55], [[a1, a3, s], [[[a1, t], 10], [[a3, t], 12], [[s, a0], 9], [[s, a2], 11], [[s, a4], 13]], 55], [[a0, a1, a3, s], [[[a0, t], 9], [[a1, t], 10], [[a3, t], 12], [[s, a2], 11], [[s, a4], 13]], 55], [[a2, a3, s], [[[a2, t], 11], [[a3, t], 12], [[s, a0], 9], [[s, a1], 10], [[s, a4], 13]], 55], [[a0, a2, a3, s], [[[a0, t], 9], [[a2, t], 11], [[a3, t], 12], [[s, a1], 10], [[s, a4], 13]], 55], [[a1, a2, a3, s], [[[a1, t], 10], [[a2, t], 11], [[a3, t], 12], [[s, a0], 9], [[s, a4], 13]], 55], [[a0, a1, a2, a3, s], [[[a0, t], 9], [[a1, t], 10], [[a2, t], 11], [[a3, t], 12], [[s, a4], 13]], 55], [[a4, s], [[[a4, t], 13], [[s, a0], 9], [[s, a1], 10], [[s, a2], 11], [[s, a3], 12]], 55], [[a0, a4, s], [[[a0, t], 9], [[a4, t], 13], [[s, a1], 10], [[s, a2], 11], [[s, a3], 12]], 55], [[a1, a4, s], [[[a1, t], 10], [[a4, t], 13], [[s, a0], 9], [[s, a2], 11], [[s, a3], 12]], 55], [[a0, a1, a4, s], [[[a0, t], 9], [[a1, t], 10], [[a4, t], 13], [[s, a2], 11], [[s, a3], 12]], 55], [[a2, a4, s], [[[a2, t], 11], [[a4, t], 13], [[s, a0], 9], [[s, a1], 10], [[s, a3], 12]], 55], [[a0, a2, a4, s], [[[a0, t], 9], [[a2, t], 11], [[a4, t], 13], [[s, a1], 10], [[s, a3], 12]], 55], [[a1, a2, a4, s], [[[a1, t], 10], [[a2, t], 11], [[a4, t], 13], [[s, a0], 9], [[s, a3], 12]], 55], [[a0, a1, a2, a4, s], [[[a0, t], 9], [[a1, t], 10], [[a2, t], 11], [[a4, t], 13], [[s, a3], 12]], 55], [[a3, a4, s], [[[a3, t], 12], [[a4, t], 13], [[s, a0], 9], [[s, a1], 10], [[s, a2], 11]], 55], [[a0, a3, a4, s], [[[a0, t], 9], [[a3, t], 12], [[a4, t], 13], [[s, a1], 10], [[s, a2], 11]], 55], [[a1, a3, a4, s], [[[a1, t], 10], [[a3, t], 12], [[a4, t], 13], [[s, a0], 9], [[s, a2], 11]], 55], [[a0, a1, a3, a4, s], [[[a0, t], 9], [[a1, t], 10], [[a3, t], 12], [[a4, t], 13], [[s, a2], 11]], 55], [[a2, a3, a4, s], [[[a2, t], 11], [[a3, t], 12], [[a4, t], 13], [[s, a0], 9], [[s, a1], 10]], 55], [[a0, a2, a3, a4, s], [[[a0, t], 9], [[a2, t], 11], [[a3, t], 12], [[a4, t], 13], [[s, a1], 10]], 55], [[a1, a2, a3, a4, s], [[[a1, t], 10], [[a2, t], 11], [[a3, t], 12], [[a4, t], 13], [[s, a0], 9]], 55], [[a0, a1, a2, a3, a4, s], [[[a0, t], 9], [[a1, t], 10], [[a2, t], 11], [[a3, t], 12], [[a4, t], 13]], 55]], 32, 32, every_subset_containing_s_and_excluding_t_is_enumerated_and_only_forward_edges_are_counted)
Enumerate every subset of the other vertices, with source included and sink excluded. For each cut list every outgoing edge and its capacity, then sum exactly. Retain actual and required counts; incoming edges are not charged to cut capacity. The exact result, retaining the requested order, is ([a0, a1, a2, a3, a4, s, t], s, t, [[[s], [[[s, a0], 9], [[s, a1], 10], [[s, a2], 11], [[s, a3], 12], [[s, a4], 13]], 55], [[a0, s], [[[a0, t], 9], [[s, a1], 10], [[s, a2], 11], [[s, a3], 12], [[s, a4], 13]], 55], [[a1, s], [[[a1, t], 10], [[s, a0], 9], [[s, a2], 11], [[s, a3], 12], [[s, a4], 13]], 55], [[a0, a1, s], [[[a0, t], 9], [[a1, t], 10], [[s, a2], 11], [[s, a3], 12], [[s, a4], 13]], 55], [[a2, s], [[[a2, t], 11], [[s, a0], 9], [[s, a1], 10], [[s, a3], 12], [[s, a4], 13]], 55], [[a0, a2, s], [[[a0, t], 9], [[a2, t], 11], [[s, a1], 10], [[s, a3], 12], [[s, a4], 13]], 55], [[a1, a2, s], [[[a1, t], 10], [[a2, t], 11], [[s, a0], 9], [[s, a3], 12], [[s, a4], 13]], 55], [[a0, a1, a2, s], [[[a0, t], 9], [[a1, t], 10], [[a2, t], 11], [[s, a3], 12], [[s, a4], 13]], 55], [[a3, s], [[[a3, t], 12], [[s, a0], 9], [[s, a1], 10], [[s, a2], 11], [[s, a4], 13]], 55], [[a0, a3, s], [[[a0, t], 9], [[a3, t], 12], [[s, a1], 10], [[s, a2], 11], [[s, a4], 13]], 55], [[a1, a3, s], [[[a1, t], 10], [[a3, t], 12], [[s, a0], 9], [[s, a2], 11], [[s, a4], 13]], 55], [[a0, a1, a3, s], [[[a0, t], 9], [[a1, t], 10], [[a3, t], 12], [[s, a2], 11], [[s, a4], 13]], 55], [[a2, a3, s], [[[a2, t], 11], [[a3, t], 12], [[s, a0], 9], [[s, a1], 10], [[s, a4], 13]], 55], [[a0, a2, a3, s], [[[a0, t], 9], [[a2, t], 11], [[a3, t], 12], [[s, a1], 10], [[s, a4], 13]], 55], [[a1, a2, a3, s], [[[a1, t], 10], [[a2, t], 11], [[a3, t], 12], [[s, a0], 9], [[s, a4], 13]], 55], [[a0, a1, a2, a3, s], [[[a0, t], 9], [[a1, t], 10], [[a2, t], 11], [[a3, t], 12], [[s, a4], 13]], 55], [[a4, s], [[[a4, t], 13], [[s, a0], 9], [[s, a1], 10], [[s, a2], 11], [[s, a3], 12]], 55], [[a0, a4, s], [[[a0, t], 9], [[a4, t], 13], [[s, a1], 10], [[s, a2], 11], [[s, a3], 12]], 55], [[a1, a4, s], [[[a1, t], 10], [[a4, t], 13], [[s, a0], 9], [[s, a2], 11], [[s, a3], 12]], 55], [[a0, a1, a4, s], [[[a0, t], 9], [[a1, t], 10], [[a4, t], 13], [[s, a2], 11], [[s, a3], 12]], 55], [[a2, a4, s], [[[a2, t], 11], [[a4, t], 13], [[s, a0], 9], [[s, a1], 10], [[s, a3], 12]], 55], [[a0, a2, a4, s], [[[a0, t], 9], [[a2, t], 11], [[a4, t], 13], [[s, a1], 10], [[s, a3], 12]], 55], [[a1, a2, a4, s], [[[a1, t], 10], [[a2, t], 11], [[a4, t], 13], [[s, a0], 9], [[s, a3], 12]], 55], [[a0, a1, a2, a4, s], [[[a0, t], 9], [[a1, t], 10], [[a2, t], 11], [[a4, t], 13], [[s, a3], 12]], 55], [[a3, a4, s], [[[a3, t], 12], [[a4, t], 13], [[s, a0], 9], [[s, a1], 10], [[s, a2], 11]], 55], [[a0, a3, a4, s], [[[a0, t], 9], [[a3, t], 12], [[a4, t], 13], [[s, a1], 10], [[s, a2], 11]], 55], [[a1, a3, a4, s], [[[a1, t], 10], [[a3, t], 12], [[a4, t], 13], [[s, a0], 9], [[s, a2], 11]], 55], [[a0, a1, a3, a4, s], [[[a0, t], 9], [[a1, t], 10], [[a3, t], 12], [[a4, t], 13], [[s, a2], 11]], 55], [[a2, a3, a4, s], [[[a2, t], 11], [[a3, t], 12], [[a4, t], 13], [[s, a0], 9], [[s, a1], 10]], 55], [[a0, a2, a3, a4, s], [[[a0, t], 9], [[a2, t], 11], [[a3, t], 12], [[a4, t], 13], [[s, a1], 10]], 55], [[a1, a2, a3, a4, s], [[[a1, t], 10], [[a2, t], 11], [[a3, t], 12], [[a4, t], 13], [[s, a0], 9]], 55], [[a0, a1, a2, a3, a4, s], [[[a0, t], 9], [[a1, t], 10], [[a2, t], 11], [[a3, t], 12], [[a4, t], 13]], 55]], 32, 32, every_subset_containing_s_and_excluding_t_is_enumerated_and_only_forward_edges_are_counted).
Reviewed step-by-step derivation
Max-flow min-cut theorem · qf-mod46-mfmc-complete-cut-enumeration · candidate
9. Every Source-Side Vertex Subset Defines One Directed Cut Audit
Use complete directed vertices=[s, a0, a1, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t]], source=s, sink=t, aligned nonnegative exact capacities=[10, 10, 11, 11]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (vertices, source, sink, cut_rows, cut_count, required_cut_count, certificate).
View answer and explanation
([a0, a1, s, t], s, t, [[[s], [[[s, a0], 10], [[s, a1], 11]], 21], [[a0, s], [[[a0, t], 10], [[s, a1], 11]], 21], [[a1, s], [[[a1, t], 11], [[s, a0], 10]], 21], [[a0, a1, s], [[[a0, t], 10], [[a1, t], 11]], 21]], 4, 4, every_subset_containing_s_and_excluding_t_is_enumerated_and_only_forward_edges_are_counted)
Enumerate every subset of the other vertices, with source included and sink excluded. For each cut list every outgoing edge and its capacity, then sum exactly. Retain actual and required counts; incoming edges are not charged to cut capacity. The exact result, retaining the requested order, is ([a0, a1, s, t], s, t, [[[s], [[[s, a0], 10], [[s, a1], 11]], 21], [[a0, s], [[[a0, t], 10], [[s, a1], 11]], 21], [[a1, s], [[[a1, t], 11], [[s, a0], 10]], 21], [[a0, a1, s], [[[a0, t], 10], [[a1, t], 11]], 21]], 4, 4, every_subset_containing_s_and_excluding_t_is_enumerated_and_only_forward_edges_are_counted).
Reviewed step-by-step derivation
Max-flow min-cut theorem · qf-mod46-mfmc-complete-cut-enumeration · candidate
10. Every Source-Side Vertex Subset Defines One Directed Cut Audit
Use complete directed vertices=[s, a0, a1, a2, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t], [s, a2], [a2, t], [a0, s]], source=s, sink=t, aligned nonnegative exact capacities=[11, 11, 12, 12, 13, 13, 16]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (vertices, source, sink, cut_rows, cut_count, required_cut_count, certificate).
View answer and explanation
([a0, a1, a2, s, t], s, t, [[[s], [[[s, a0], 11], [[s, a1], 12], [[s, a2], 13]], 36], [[a0, s], [[[a0, t], 11], [[s, a1], 12], [[s, a2], 13]], 36], [[a1, s], [[[a1, t], 12], [[s, a0], 11], [[s, a2], 13]], 36], [[a0, a1, s], [[[a0, t], 11], [[a1, t], 12], [[s, a2], 13]], 36], [[a2, s], [[[a2, t], 13], [[s, a0], 11], [[s, a1], 12]], 36], [[a0, a2, s], [[[a0, t], 11], [[a2, t], 13], [[s, a1], 12]], 36], [[a1, a2, s], [[[a1, t], 12], [[a2, t], 13], [[s, a0], 11]], 36], [[a0, a1, a2, s], [[[a0, t], 11], [[a1, t], 12], [[a2, t], 13]], 36]], 8, 8, every_subset_containing_s_and_excluding_t_is_enumerated_and_only_forward_edges_are_counted)
Enumerate every subset of the other vertices, with source included and sink excluded. For each cut list every outgoing edge and its capacity, then sum exactly. Retain actual and required counts; incoming edges are not charged to cut capacity. The exact result, retaining the requested order, is ([a0, a1, a2, s, t], s, t, [[[s], [[[s, a0], 11], [[s, a1], 12], [[s, a2], 13]], 36], [[a0, s], [[[a0, t], 11], [[s, a1], 12], [[s, a2], 13]], 36], [[a1, s], [[[a1, t], 12], [[s, a0], 11], [[s, a2], 13]], 36], [[a0, a1, s], [[[a0, t], 11], [[a1, t], 12], [[s, a2], 13]], 36], [[a2, s], [[[a2, t], 13], [[s, a0], 11], [[s, a1], 12]], 36], [[a0, a2, s], [[[a0, t], 11], [[a2, t], 13], [[s, a1], 12]], 36], [[a1, a2, s], [[[a1, t], 12], [[a2, t], 13], [[s, a0], 11]], 36], [[a0, a1, a2, s], [[[a0, t], 11], [[a1, t], 12], [[a2, t], 13]], 36]], 8, 8, every_subset_containing_s_and_excluding_t_is_enumerated_and_only_forward_edges_are_counted).
Reviewed step-by-step derivation
Max-flow min-cut theorem · qf-mod46-mfmc-complete-cut-enumeration · candidate
11. Every Source-Side Vertex Subset Defines One Directed Cut Audit
Use complete directed vertices=[s, a0, a1, a2, a3, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t], [s, a2], [a2, t], [s, a3], [a3, t]], source=s, sink=t, aligned nonnegative exact capacities=[12, 12, 13, 13, 14, 14, 15, 15]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (vertices, source, sink, cut_rows, cut_count, required_cut_count, certificate).
View answer and explanation
([a0, a1, a2, a3, s, t], s, t, [[[s], [[[s, a0], 12], [[s, a1], 13], [[s, a2], 14], [[s, a3], 15]], 54], [[a0, s], [[[a0, t], 12], [[s, a1], 13], [[s, a2], 14], [[s, a3], 15]], 54], [[a1, s], [[[a1, t], 13], [[s, a0], 12], [[s, a2], 14], [[s, a3], 15]], 54], [[a0, a1, s], [[[a0, t], 12], [[a1, t], 13], [[s, a2], 14], [[s, a3], 15]], 54], [[a2, s], [[[a2, t], 14], [[s, a0], 12], [[s, a1], 13], [[s, a3], 15]], 54], [[a0, a2, s], [[[a0, t], 12], [[a2, t], 14], [[s, a1], 13], [[s, a3], 15]], 54], [[a1, a2, s], [[[a1, t], 13], [[a2, t], 14], [[s, a0], 12], [[s, a3], 15]], 54], [[a0, a1, a2, s], [[[a0, t], 12], [[a1, t], 13], [[a2, t], 14], [[s, a3], 15]], 54], [[a3, s], [[[a3, t], 15], [[s, a0], 12], [[s, a1], 13], [[s, a2], 14]], 54], [[a0, a3, s], [[[a0, t], 12], [[a3, t], 15], [[s, a1], 13], [[s, a2], 14]], 54], [[a1, a3, s], [[[a1, t], 13], [[a3, t], 15], [[s, a0], 12], [[s, a2], 14]], 54], [[a0, a1, a3, s], [[[a0, t], 12], [[a1, t], 13], [[a3, t], 15], [[s, a2], 14]], 54], [[a2, a3, s], [[[a2, t], 14], [[a3, t], 15], [[s, a0], 12], [[s, a1], 13]], 54], [[a0, a2, a3, s], [[[a0, t], 12], [[a2, t], 14], [[a3, t], 15], [[s, a1], 13]], 54], [[a1, a2, a3, s], [[[a1, t], 13], [[a2, t], 14], [[a3, t], 15], [[s, a0], 12]], 54], [[a0, a1, a2, a3, s], [[[a0, t], 12], [[a1, t], 13], [[a2, t], 14], [[a3, t], 15]], 54]], 16, 16, every_subset_containing_s_and_excluding_t_is_enumerated_and_only_forward_edges_are_counted)
Enumerate every subset of the other vertices, with source included and sink excluded. For each cut list every outgoing edge and its capacity, then sum exactly. Retain actual and required counts; incoming edges are not charged to cut capacity. The exact result, retaining the requested order, is ([a0, a1, a2, a3, s, t], s, t, [[[s], [[[s, a0], 12], [[s, a1], 13], [[s, a2], 14], [[s, a3], 15]], 54], [[a0, s], [[[a0, t], 12], [[s, a1], 13], [[s, a2], 14], [[s, a3], 15]], 54], [[a1, s], [[[a1, t], 13], [[s, a0], 12], [[s, a2], 14], [[s, a3], 15]], 54], [[a0, a1, s], [[[a0, t], 12], [[a1, t], 13], [[s, a2], 14], [[s, a3], 15]], 54], [[a2, s], [[[a2, t], 14], [[s, a0], 12], [[s, a1], 13], [[s, a3], 15]], 54], [[a0, a2, s], [[[a0, t], 12], [[a2, t], 14], [[s, a1], 13], [[s, a3], 15]], 54], [[a1, a2, s], [[[a1, t], 13], [[a2, t], 14], [[s, a0], 12], [[s, a3], 15]], 54], [[a0, a1, a2, s], [[[a0, t], 12], [[a1, t], 13], [[a2, t], 14], [[s, a3], 15]], 54], [[a3, s], [[[a3, t], 15], [[s, a0], 12], [[s, a1], 13], [[s, a2], 14]], 54], [[a0, a3, s], [[[a0, t], 12], [[a3, t], 15], [[s, a1], 13], [[s, a2], 14]], 54], [[a1, a3, s], [[[a1, t], 13], [[a3, t], 15], [[s, a0], 12], [[s, a2], 14]], 54], [[a0, a1, a3, s], [[[a0, t], 12], [[a1, t], 13], [[a3, t], 15], [[s, a2], 14]], 54], [[a2, a3, s], [[[a2, t], 14], [[a3, t], 15], [[s, a0], 12], [[s, a1], 13]], 54], [[a0, a2, a3, s], [[[a0, t], 12], [[a2, t], 14], [[a3, t], 15], [[s, a1], 13]], 54], [[a1, a2, a3, s], [[[a1, t], 13], [[a2, t], 14], [[a3, t], 15], [[s, a0], 12]], 54], [[a0, a1, a2, a3, s], [[[a0, t], 12], [[a1, t], 13], [[a2, t], 14], [[a3, t], 15]], 54]], 16, 16, every_subset_containing_s_and_excluding_t_is_enumerated_and_only_forward_edges_are_counted).
Reviewed step-by-step derivation
Max-flow min-cut theorem · qf-mod46-mfmc-complete-cut-enumeration · candidate
12. Every Source-Side Vertex Subset Defines One Directed Cut Audit
Use complete directed vertices=[s, a0, a1, a2, a3, a4, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t], [s, a2], [a2, t], [s, a3], [a3, t], [s, a4], [a4, t]], source=s, sink=t, aligned nonnegative exact capacities=[13, 13, 14, 14, 15, 15, 16, 16, 17, 17]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (vertices, source, sink, cut_rows, cut_count, required_cut_count, certificate).
View answer and explanation
([a0, a1, a2, a3, a4, s, t], s, t, [[[s], [[[s, a0], 13], [[s, a1], 14], [[s, a2], 15], [[s, a3], 16], [[s, a4], 17]], 75], [[a0, s], [[[a0, t], 13], [[s, a1], 14], [[s, a2], 15], [[s, a3], 16], [[s, a4], 17]], 75], [[a1, s], [[[a1, t], 14], [[s, a0], 13], [[s, a2], 15], [[s, a3], 16], [[s, a4], 17]], 75], [[a0, a1, s], [[[a0, t], 13], [[a1, t], 14], [[s, a2], 15], [[s, a3], 16], [[s, a4], 17]], 75], [[a2, s], [[[a2, t], 15], [[s, a0], 13], [[s, a1], 14], [[s, a3], 16], [[s, a4], 17]], 75], [[a0, a2, s], [[[a0, t], 13], [[a2, t], 15], [[s, a1], 14], [[s, a3], 16], [[s, a4], 17]], 75], [[a1, a2, s], [[[a1, t], 14], [[a2, t], 15], [[s, a0], 13], [[s, a3], 16], [[s, a4], 17]], 75], [[a0, a1, a2, s], [[[a0, t], 13], [[a1, t], 14], [[a2, t], 15], [[s, a3], 16], [[s, a4], 17]], 75], [[a3, s], [[[a3, t], 16], [[s, a0], 13], [[s, a1], 14], [[s, a2], 15], [[s, a4], 17]], 75], [[a0, a3, s], [[[a0, t], 13], [[a3, t], 16], [[s, a1], 14], [[s, a2], 15], [[s, a4], 17]], 75], [[a1, a3, s], [[[a1, t], 14], [[a3, t], 16], [[s, a0], 13], [[s, a2], 15], [[s, a4], 17]], 75], [[a0, a1, a3, s], [[[a0, t], 13], [[a1, t], 14], [[a3, t], 16], [[s, a2], 15], [[s, a4], 17]], 75], [[a2, a3, s], [[[a2, t], 15], [[a3, t], 16], [[s, a0], 13], [[s, a1], 14], [[s, a4], 17]], 75], [[a0, a2, a3, s], [[[a0, t], 13], [[a2, t], 15], [[a3, t], 16], [[s, a1], 14], [[s, a4], 17]], 75], [[a1, a2, a3, s], [[[a1, t], 14], [[a2, t], 15], [[a3, t], 16], [[s, a0], 13], [[s, a4], 17]], 75], [[a0, a1, a2, a3, s], [[[a0, t], 13], [[a1, t], 14], [[a2, t], 15], [[a3, t], 16], [[s, a4], 17]], 75], [[a4, s], [[[a4, t], 17], [[s, a0], 13], [[s, a1], 14], [[s, a2], 15], [[s, a3], 16]], 75], [[a0, a4, s], [[[a0, t], 13], [[a4, t], 17], [[s, a1], 14], [[s, a2], 15], [[s, a3], 16]], 75], [[a1, a4, s], [[[a1, t], 14], [[a4, t], 17], [[s, a0], 13], [[s, a2], 15], [[s, a3], 16]], 75], [[a0, a1, a4, s], [[[a0, t], 13], [[a1, t], 14], [[a4, t], 17], [[s, a2], 15], [[s, a3], 16]], 75], [[a2, a4, s], [[[a2, t], 15], [[a4, t], 17], [[s, a0], 13], [[s, a1], 14], [[s, a3], 16]], 75], [[a0, a2, a4, s], [[[a0, t], 13], [[a2, t], 15], [[a4, t], 17], [[s, a1], 14], [[s, a3], 16]], 75], [[a1, a2, a4, s], [[[a1, t], 14], [[a2, t], 15], [[a4, t], 17], [[s, a0], 13], [[s, a3], 16]], 75], [[a0, a1, a2, a4, s], [[[a0, t], 13], [[a1, t], 14], [[a2, t], 15], [[a4, t], 17], [[s, a3], 16]], 75], [[a3, a4, s], [[[a3, t], 16], [[a4, t], 17], [[s, a0], 13], [[s, a1], 14], [[s, a2], 15]], 75], [[a0, a3, a4, s], [[[a0, t], 13], [[a3, t], 16], [[a4, t], 17], [[s, a1], 14], [[s, a2], 15]], 75], [[a1, a3, a4, s], [[[a1, t], 14], [[a3, t], 16], [[a4, t], 17], [[s, a0], 13], [[s, a2], 15]], 75], [[a0, a1, a3, a4, s], [[[a0, t], 13], [[a1, t], 14], [[a3, t], 16], [[a4, t], 17], [[s, a2], 15]], 75], [[a2, a3, a4, s], [[[a2, t], 15], [[a3, t], 16], [[a4, t], 17], [[s, a0], 13], [[s, a1], 14]], 75], [[a0, a2, a3, a4, s], [[[a0, t], 13], [[a2, t], 15], [[a3, t], 16], [[a4, t], 17], [[s, a1], 14]], 75], [[a1, a2, a3, a4, s], [[[a1, t], 14], [[a2, t], 15], [[a3, t], 16], [[a4, t], 17], [[s, a0], 13]], 75], [[a0, a1, a2, a3, a4, s], [[[a0, t], 13], [[a1, t], 14], [[a2, t], 15], [[a3, t], 16], [[a4, t], 17]], 75]], 32, 32, every_subset_containing_s_and_excluding_t_is_enumerated_and_only_forward_edges_are_counted)
Enumerate every subset of the other vertices, with source included and sink excluded. For each cut list every outgoing edge and its capacity, then sum exactly. Retain actual and required counts; incoming edges are not charged to cut capacity. The exact result, retaining the requested order, is ([a0, a1, a2, a3, a4, s, t], s, t, [[[s], [[[s, a0], 13], [[s, a1], 14], [[s, a2], 15], [[s, a3], 16], [[s, a4], 17]], 75], [[a0, s], [[[a0, t], 13], [[s, a1], 14], [[s, a2], 15], [[s, a3], 16], [[s, a4], 17]], 75], [[a1, s], [[[a1, t], 14], [[s, a0], 13], [[s, a2], 15], [[s, a3], 16], [[s, a4], 17]], 75], [[a0, a1, s], [[[a0, t], 13], [[a1, t], 14], [[s, a2], 15], [[s, a3], 16], [[s, a4], 17]], 75], [[a2, s], [[[a2, t], 15], [[s, a0], 13], [[s, a1], 14], [[s, a3], 16], [[s, a4], 17]], 75], [[a0, a2, s], [[[a0, t], 13], [[a2, t], 15], [[s, a1], 14], [[s, a3], 16], [[s, a4], 17]], 75], [[a1, a2, s], [[[a1, t], 14], [[a2, t], 15], [[s, a0], 13], [[s, a3], 16], [[s, a4], 17]], 75], [[a0, a1, a2, s], [[[a0, t], 13], [[a1, t], 14], [[a2, t], 15], [[s, a3], 16], [[s, a4], 17]], 75], [[a3, s], [[[a3, t], 16], [[s, a0], 13], [[s, a1], 14], [[s, a2], 15], [[s, a4], 17]], 75], [[a0, a3, s], [[[a0, t], 13], [[a3, t], 16], [[s, a1], 14], [[s, a2], 15], [[s, a4], 17]], 75], [[a1, a3, s], [[[a1, t], 14], [[a3, t], 16], [[s, a0], 13], [[s, a2], 15], [[s, a4], 17]], 75], [[a0, a1, a3, s], [[[a0, t], 13], [[a1, t], 14], [[a3, t], 16], [[s, a2], 15], [[s, a4], 17]], 75], [[a2, a3, s], [[[a2, t], 15], [[a3, t], 16], [[s, a0], 13], [[s, a1], 14], [[s, a4], 17]], 75], [[a0, a2, a3, s], [[[a0, t], 13], [[a2, t], 15], [[a3, t], 16], [[s, a1], 14], [[s, a4], 17]], 75], [[a1, a2, a3, s], [[[a1, t], 14], [[a2, t], 15], [[a3, t], 16], [[s, a0], 13], [[s, a4], 17]], 75], [[a0, a1, a2, a3, s], [[[a0, t], 13], [[a1, t], 14], [[a2, t], 15], [[a3, t], 16], [[s, a4], 17]], 75], [[a4, s], [[[a4, t], 17], [[s, a0], 13], [[s, a1], 14], [[s, a2], 15], [[s, a3], 16]], 75], [[a0, a4, s], [[[a0, t], 13], [[a4, t], 17], [[s, a1], 14], [[s, a2], 15], [[s, a3], 16]], 75], [[a1, a4, s], [[[a1, t], 14], [[a4, t], 17], [[s, a0], 13], [[s, a2], 15], [[s, a3], 16]], 75], [[a0, a1, a4, s], [[[a0, t], 13], [[a1, t], 14], [[a4, t], 17], [[s, a2], 15], [[s, a3], 16]], 75], [[a2, a4, s], [[[a2, t], 15], [[a4, t], 17], [[s, a0], 13], [[s, a1], 14], [[s, a3], 16]], 75], [[a0, a2, a4, s], [[[a0, t], 13], [[a2, t], 15], [[a4, t], 17], [[s, a1], 14], [[s, a3], 16]], 75], [[a1, a2, a4, s], [[[a1, t], 14], [[a2, t], 15], [[a4, t], 17], [[s, a0], 13], [[s, a3], 16]], 75], [[a0, a1, a2, a4, s], [[[a0, t], 13], [[a1, t], 14], [[a2, t], 15], [[a4, t], 17], [[s, a3], 16]], 75], [[a3, a4, s], [[[a3, t], 16], [[a4, t], 17], [[s, a0], 13], [[s, a1], 14], [[s, a2], 15]], 75], [[a0, a3, a4, s], [[[a0, t], 13], [[a3, t], 16], [[a4, t], 17], [[s, a1], 14], [[s, a2], 15]], 75], [[a1, a3, a4, s], [[[a1, t], 14], [[a3, t], 16], [[a4, t], 17], [[s, a0], 13], [[s, a2], 15]], 75], [[a0, a1, a3, a4, s], [[[a0, t], 13], [[a1, t], 14], [[a3, t], 16], [[a4, t], 17], [[s, a2], 15]], 75], [[a2, a3, a4, s], [[[a2, t], 15], [[a3, t], 16], [[a4, t], 17], [[s, a0], 13], [[s, a1], 14]], 75], [[a0, a2, a3, a4, s], [[[a0, t], 13], [[a2, t], 15], [[a3, t], 16], [[a4, t], 17], [[s, a1], 14]], 75], [[a1, a2, a3, a4, s], [[[a1, t], 14], [[a2, t], 15], [[a3, t], 16], [[a4, t], 17], [[s, a0], 13]], 75], [[a0, a1, a2, a3, a4, s], [[[a0, t], 13], [[a1, t], 14], [[a2, t], 15], [[a3, t], 16], [[a4, t], 17]], 75]], 32, 32, every_subset_containing_s_and_excluding_t_is_enumerated_and_only_forward_edges_are_counted).
Reviewed step-by-step derivation
Max-flow min-cut theorem · qf-mod46-mfmc-complete-cut-enumeration · candidate
13. Every Source-Side Vertex Subset Defines One Directed Cut Audit
Use complete directed vertices=[s, a0, a1, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t], [a0, s]], source=s, sink=t, aligned nonnegative exact capacities=[14, 14, 15, 15, 19]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (vertices, source, sink, cut_rows, cut_count, required_cut_count, certificate).
View answer and explanation
([a0, a1, s, t], s, t, [[[s], [[[s, a0], 14], [[s, a1], 15]], 29], [[a0, s], [[[a0, t], 14], [[s, a1], 15]], 29], [[a1, s], [[[a1, t], 15], [[s, a0], 14]], 29], [[a0, a1, s], [[[a0, t], 14], [[a1, t], 15]], 29]], 4, 4, every_subset_containing_s_and_excluding_t_is_enumerated_and_only_forward_edges_are_counted)
Enumerate every subset of the other vertices, with source included and sink excluded. For each cut list every outgoing edge and its capacity, then sum exactly. Retain actual and required counts; incoming edges are not charged to cut capacity. The exact result, retaining the requested order, is ([a0, a1, s, t], s, t, [[[s], [[[s, a0], 14], [[s, a1], 15]], 29], [[a0, s], [[[a0, t], 14], [[s, a1], 15]], 29], [[a1, s], [[[a1, t], 15], [[s, a0], 14]], 29], [[a0, a1, s], [[[a0, t], 14], [[a1, t], 15]], 29]], 4, 4, every_subset_containing_s_and_excluding_t_is_enumerated_and_only_forward_edges_are_counted).
Reviewed step-by-step derivation
Max-flow min-cut theorem · qf-mod46-mfmc-complete-cut-enumeration · candidate
14. Every Source-Side Vertex Subset Defines One Directed Cut Audit
Use complete directed vertices=[s, a0, a1, a2, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t], [s, a2], [a2, t]], source=s, sink=t, aligned nonnegative exact capacities=[15, 15, 16, 16, 17, 17]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (vertices, source, sink, cut_rows, cut_count, required_cut_count, certificate).
View answer and explanation
([a0, a1, a2, s, t], s, t, [[[s], [[[s, a0], 15], [[s, a1], 16], [[s, a2], 17]], 48], [[a0, s], [[[a0, t], 15], [[s, a1], 16], [[s, a2], 17]], 48], [[a1, s], [[[a1, t], 16], [[s, a0], 15], [[s, a2], 17]], 48], [[a0, a1, s], [[[a0, t], 15], [[a1, t], 16], [[s, a2], 17]], 48], [[a2, s], [[[a2, t], 17], [[s, a0], 15], [[s, a1], 16]], 48], [[a0, a2, s], [[[a0, t], 15], [[a2, t], 17], [[s, a1], 16]], 48], [[a1, a2, s], [[[a1, t], 16], [[a2, t], 17], [[s, a0], 15]], 48], [[a0, a1, a2, s], [[[a0, t], 15], [[a1, t], 16], [[a2, t], 17]], 48]], 8, 8, every_subset_containing_s_and_excluding_t_is_enumerated_and_only_forward_edges_are_counted)
Enumerate every subset of the other vertices, with source included and sink excluded. For each cut list every outgoing edge and its capacity, then sum exactly. Retain actual and required counts; incoming edges are not charged to cut capacity. The exact result, retaining the requested order, is ([a0, a1, a2, s, t], s, t, [[[s], [[[s, a0], 15], [[s, a1], 16], [[s, a2], 17]], 48], [[a0, s], [[[a0, t], 15], [[s, a1], 16], [[s, a2], 17]], 48], [[a1, s], [[[a1, t], 16], [[s, a0], 15], [[s, a2], 17]], 48], [[a0, a1, s], [[[a0, t], 15], [[a1, t], 16], [[s, a2], 17]], 48], [[a2, s], [[[a2, t], 17], [[s, a0], 15], [[s, a1], 16]], 48], [[a0, a2, s], [[[a0, t], 15], [[a2, t], 17], [[s, a1], 16]], 48], [[a1, a2, s], [[[a1, t], 16], [[a2, t], 17], [[s, a0], 15]], 48], [[a0, a1, a2, s], [[[a0, t], 15], [[a1, t], 16], [[a2, t], 17]], 48]], 8, 8, every_subset_containing_s_and_excluding_t_is_enumerated_and_only_forward_edges_are_counted).
Reviewed step-by-step derivation
Max-flow min-cut theorem · qf-mod46-mfmc-complete-cut-enumeration · candidate
15. Every Source-Side Vertex Subset Defines One Directed Cut Audit
Use complete directed vertices=[s, a0, a1, a2, a3, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t], [s, a2], [a2, t], [s, a3], [a3, t]], source=s, sink=t, aligned nonnegative exact capacities=[16, 16, 17, 17, 18, 18, 19, 19]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (vertices, source, sink, cut_rows, cut_count, required_cut_count, certificate).
View answer and explanation
([a0, a1, a2, a3, s, t], s, t, [[[s], [[[s, a0], 16], [[s, a1], 17], [[s, a2], 18], [[s, a3], 19]], 70], [[a0, s], [[[a0, t], 16], [[s, a1], 17], [[s, a2], 18], [[s, a3], 19]], 70], [[a1, s], [[[a1, t], 17], [[s, a0], 16], [[s, a2], 18], [[s, a3], 19]], 70], [[a0, a1, s], [[[a0, t], 16], [[a1, t], 17], [[s, a2], 18], [[s, a3], 19]], 70], [[a2, s], [[[a2, t], 18], [[s, a0], 16], [[s, a1], 17], [[s, a3], 19]], 70], [[a0, a2, s], [[[a0, t], 16], [[a2, t], 18], [[s, a1], 17], [[s, a3], 19]], 70], [[a1, a2, s], [[[a1, t], 17], [[a2, t], 18], [[s, a0], 16], [[s, a3], 19]], 70], [[a0, a1, a2, s], [[[a0, t], 16], [[a1, t], 17], [[a2, t], 18], [[s, a3], 19]], 70], [[a3, s], [[[a3, t], 19], [[s, a0], 16], [[s, a1], 17], [[s, a2], 18]], 70], [[a0, a3, s], [[[a0, t], 16], [[a3, t], 19], [[s, a1], 17], [[s, a2], 18]], 70], [[a1, a3, s], [[[a1, t], 17], [[a3, t], 19], [[s, a0], 16], [[s, a2], 18]], 70], [[a0, a1, a3, s], [[[a0, t], 16], [[a1, t], 17], [[a3, t], 19], [[s, a2], 18]], 70], [[a2, a3, s], [[[a2, t], 18], [[a3, t], 19], [[s, a0], 16], [[s, a1], 17]], 70], [[a0, a2, a3, s], [[[a0, t], 16], [[a2, t], 18], [[a3, t], 19], [[s, a1], 17]], 70], [[a1, a2, a3, s], [[[a1, t], 17], [[a2, t], 18], [[a3, t], 19], [[s, a0], 16]], 70], [[a0, a1, a2, a3, s], [[[a0, t], 16], [[a1, t], 17], [[a2, t], 18], [[a3, t], 19]], 70]], 16, 16, every_subset_containing_s_and_excluding_t_is_enumerated_and_only_forward_edges_are_counted)
Enumerate every subset of the other vertices, with source included and sink excluded. For each cut list every outgoing edge and its capacity, then sum exactly. Retain actual and required counts; incoming edges are not charged to cut capacity. The exact result, retaining the requested order, is ([a0, a1, a2, a3, s, t], s, t, [[[s], [[[s, a0], 16], [[s, a1], 17], [[s, a2], 18], [[s, a3], 19]], 70], [[a0, s], [[[a0, t], 16], [[s, a1], 17], [[s, a2], 18], [[s, a3], 19]], 70], [[a1, s], [[[a1, t], 17], [[s, a0], 16], [[s, a2], 18], [[s, a3], 19]], 70], [[a0, a1, s], [[[a0, t], 16], [[a1, t], 17], [[s, a2], 18], [[s, a3], 19]], 70], [[a2, s], [[[a2, t], 18], [[s, a0], 16], [[s, a1], 17], [[s, a3], 19]], 70], [[a0, a2, s], [[[a0, t], 16], [[a2, t], 18], [[s, a1], 17], [[s, a3], 19]], 70], [[a1, a2, s], [[[a1, t], 17], [[a2, t], 18], [[s, a0], 16], [[s, a3], 19]], 70], [[a0, a1, a2, s], [[[a0, t], 16], [[a1, t], 17], [[a2, t], 18], [[s, a3], 19]], 70], [[a3, s], [[[a3, t], 19], [[s, a0], 16], [[s, a1], 17], [[s, a2], 18]], 70], [[a0, a3, s], [[[a0, t], 16], [[a3, t], 19], [[s, a1], 17], [[s, a2], 18]], 70], [[a1, a3, s], [[[a1, t], 17], [[a3, t], 19], [[s, a0], 16], [[s, a2], 18]], 70], [[a0, a1, a3, s], [[[a0, t], 16], [[a1, t], 17], [[a3, t], 19], [[s, a2], 18]], 70], [[a2, a3, s], [[[a2, t], 18], [[a3, t], 19], [[s, a0], 16], [[s, a1], 17]], 70], [[a0, a2, a3, s], [[[a0, t], 16], [[a2, t], 18], [[a3, t], 19], [[s, a1], 17]], 70], [[a1, a2, a3, s], [[[a1, t], 17], [[a2, t], 18], [[a3, t], 19], [[s, a0], 16]], 70], [[a0, a1, a2, a3, s], [[[a0, t], 16], [[a1, t], 17], [[a2, t], 18], [[a3, t], 19]], 70]], 16, 16, every_subset_containing_s_and_excluding_t_is_enumerated_and_only_forward_edges_are_counted).
Reviewed step-by-step derivation
Max-flow min-cut theorem · qf-mod46-mfmc-complete-cut-enumeration · candidate
16. Every Source-Side Vertex Subset Defines One Directed Cut Audit
Use complete directed vertices=[s, a0, a1, a2, a3, a4, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t], [s, a2], [a2, t], [s, a3], [a3, t], [s, a4], [a4, t], [a0, s]], source=s, sink=t, aligned nonnegative exact capacities=[17, 17, 18, 18, 19, 19, 20, 20, 21, 21, 22]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (vertices, source, sink, cut_rows, cut_count, required_cut_count, certificate).
View answer and explanation
([a0, a1, a2, a3, a4, s, t], s, t, [[[s], [[[s, a0], 17], [[s, a1], 18], [[s, a2], 19], [[s, a3], 20], [[s, a4], 21]], 95], [[a0, s], [[[a0, t], 17], [[s, a1], 18], [[s, a2], 19], [[s, a3], 20], [[s, a4], 21]], 95], [[a1, s], [[[a1, t], 18], [[s, a0], 17], [[s, a2], 19], [[s, a3], 20], [[s, a4], 21]], 95], [[a0, a1, s], [[[a0, t], 17], [[a1, t], 18], [[s, a2], 19], [[s, a3], 20], [[s, a4], 21]], 95], [[a2, s], [[[a2, t], 19], [[s, a0], 17], [[s, a1], 18], [[s, a3], 20], [[s, a4], 21]], 95], [[a0, a2, s], [[[a0, t], 17], [[a2, t], 19], [[s, a1], 18], [[s, a3], 20], [[s, a4], 21]], 95], [[a1, a2, s], [[[a1, t], 18], [[a2, t], 19], [[s, a0], 17], [[s, a3], 20], [[s, a4], 21]], 95], [[a0, a1, a2, s], [[[a0, t], 17], [[a1, t], 18], [[a2, t], 19], [[s, a3], 20], [[s, a4], 21]], 95], [[a3, s], [[[a3, t], 20], [[s, a0], 17], [[s, a1], 18], [[s, a2], 19], [[s, a4], 21]], 95], [[a0, a3, s], [[[a0, t], 17], [[a3, t], 20], [[s, a1], 18], [[s, a2], 19], [[s, a4], 21]], 95], [[a1, a3, s], [[[a1, t], 18], [[a3, t], 20], [[s, a0], 17], [[s, a2], 19], [[s, a4], 21]], 95], [[a0, a1, a3, s], [[[a0, t], 17], [[a1, t], 18], [[a3, t], 20], [[s, a2], 19], [[s, a4], 21]], 95], [[a2, a3, s], [[[a2, t], 19], [[a3, t], 20], [[s, a0], 17], [[s, a1], 18], [[s, a4], 21]], 95], [[a0, a2, a3, s], [[[a0, t], 17], [[a2, t], 19], [[a3, t], 20], [[s, a1], 18], [[s, a4], 21]], 95], [[a1, a2, a3, s], [[[a1, t], 18], [[a2, t], 19], [[a3, t], 20], [[s, a0], 17], [[s, a4], 21]], 95], [[a0, a1, a2, a3, s], [[[a0, t], 17], [[a1, t], 18], [[a2, t], 19], [[a3, t], 20], [[s, a4], 21]], 95], [[a4, s], [[[a4, t], 21], [[s, a0], 17], [[s, a1], 18], [[s, a2], 19], [[s, a3], 20]], 95], [[a0, a4, s], [[[a0, t], 17], [[a4, t], 21], [[s, a1], 18], [[s, a2], 19], [[s, a3], 20]], 95], [[a1, a4, s], [[[a1, t], 18], [[a4, t], 21], [[s, a0], 17], [[s, a2], 19], [[s, a3], 20]], 95], [[a0, a1, a4, s], [[[a0, t], 17], [[a1, t], 18], [[a4, t], 21], [[s, a2], 19], [[s, a3], 20]], 95], [[a2, a4, s], [[[a2, t], 19], [[a4, t], 21], [[s, a0], 17], [[s, a1], 18], [[s, a3], 20]], 95], [[a0, a2, a4, s], [[[a0, t], 17], [[a2, t], 19], [[a4, t], 21], [[s, a1], 18], [[s, a3], 20]], 95], [[a1, a2, a4, s], [[[a1, t], 18], [[a2, t], 19], [[a4, t], 21], [[s, a0], 17], [[s, a3], 20]], 95], [[a0, a1, a2, a4, s], [[[a0, t], 17], [[a1, t], 18], [[a2, t], 19], [[a4, t], 21], [[s, a3], 20]], 95], [[a3, a4, s], [[[a3, t], 20], [[a4, t], 21], [[s, a0], 17], [[s, a1], 18], [[s, a2], 19]], 95], [[a0, a3, a4, s], [[[a0, t], 17], [[a3, t], 20], [[a4, t], 21], [[s, a1], 18], [[s, a2], 19]], 95], [[a1, a3, a4, s], [[[a1, t], 18], [[a3, t], 20], [[a4, t], 21], [[s, a0], 17], [[s, a2], 19]], 95], [[a0, a1, a3, a4, s], [[[a0, t], 17], [[a1, t], 18], [[a3, t], 20], [[a4, t], 21], [[s, a2], 19]], 95], [[a2, a3, a4, s], [[[a2, t], 19], [[a3, t], 20], [[a4, t], 21], [[s, a0], 17], [[s, a1], 18]], 95], [[a0, a2, a3, a4, s], [[[a0, t], 17], [[a2, t], 19], [[a3, t], 20], [[a4, t], 21], [[s, a1], 18]], 95], [[a1, a2, a3, a4, s], [[[a1, t], 18], [[a2, t], 19], [[a3, t], 20], [[a4, t], 21], [[s, a0], 17]], 95], [[a0, a1, a2, a3, a4, s], [[[a0, t], 17], [[a1, t], 18], [[a2, t], 19], [[a3, t], 20], [[a4, t], 21]], 95]], 32, 32, every_subset_containing_s_and_excluding_t_is_enumerated_and_only_forward_edges_are_counted)
Enumerate every subset of the other vertices, with source included and sink excluded. For each cut list every outgoing edge and its capacity, then sum exactly. Retain actual and required counts; incoming edges are not charged to cut capacity. The exact result, retaining the requested order, is ([a0, a1, a2, a3, a4, s, t], s, t, [[[s], [[[s, a0], 17], [[s, a1], 18], [[s, a2], 19], [[s, a3], 20], [[s, a4], 21]], 95], [[a0, s], [[[a0, t], 17], [[s, a1], 18], [[s, a2], 19], [[s, a3], 20], [[s, a4], 21]], 95], [[a1, s], [[[a1, t], 18], [[s, a0], 17], [[s, a2], 19], [[s, a3], 20], [[s, a4], 21]], 95], [[a0, a1, s], [[[a0, t], 17], [[a1, t], 18], [[s, a2], 19], [[s, a3], 20], [[s, a4], 21]], 95], [[a2, s], [[[a2, t], 19], [[s, a0], 17], [[s, a1], 18], [[s, a3], 20], [[s, a4], 21]], 95], [[a0, a2, s], [[[a0, t], 17], [[a2, t], 19], [[s, a1], 18], [[s, a3], 20], [[s, a4], 21]], 95], [[a1, a2, s], [[[a1, t], 18], [[a2, t], 19], [[s, a0], 17], [[s, a3], 20], [[s, a4], 21]], 95], [[a0, a1, a2, s], [[[a0, t], 17], [[a1, t], 18], [[a2, t], 19], [[s, a3], 20], [[s, a4], 21]], 95], [[a3, s], [[[a3, t], 20], [[s, a0], 17], [[s, a1], 18], [[s, a2], 19], [[s, a4], 21]], 95], [[a0, a3, s], [[[a0, t], 17], [[a3, t], 20], [[s, a1], 18], [[s, a2], 19], [[s, a4], 21]], 95], [[a1, a3, s], [[[a1, t], 18], [[a3, t], 20], [[s, a0], 17], [[s, a2], 19], [[s, a4], 21]], 95], [[a0, a1, a3, s], [[[a0, t], 17], [[a1, t], 18], [[a3, t], 20], [[s, a2], 19], [[s, a4], 21]], 95], [[a2, a3, s], [[[a2, t], 19], [[a3, t], 20], [[s, a0], 17], [[s, a1], 18], [[s, a4], 21]], 95], [[a0, a2, a3, s], [[[a0, t], 17], [[a2, t], 19], [[a3, t], 20], [[s, a1], 18], [[s, a4], 21]], 95], [[a1, a2, a3, s], [[[a1, t], 18], [[a2, t], 19], [[a3, t], 20], [[s, a0], 17], [[s, a4], 21]], 95], [[a0, a1, a2, a3, s], [[[a0, t], 17], [[a1, t], 18], [[a2, t], 19], [[a3, t], 20], [[s, a4], 21]], 95], [[a4, s], [[[a4, t], 21], [[s, a0], 17], [[s, a1], 18], [[s, a2], 19], [[s, a3], 20]], 95], [[a0, a4, s], [[[a0, t], 17], [[a4, t], 21], [[s, a1], 18], [[s, a2], 19], [[s, a3], 20]], 95], [[a1, a4, s], [[[a1, t], 18], [[a4, t], 21], [[s, a0], 17], [[s, a2], 19], [[s, a3], 20]], 95], [[a0, a1, a4, s], [[[a0, t], 17], [[a1, t], 18], [[a4, t], 21], [[s, a2], 19], [[s, a3], 20]], 95], [[a2, a4, s], [[[a2, t], 19], [[a4, t], 21], [[s, a0], 17], [[s, a1], 18], [[s, a3], 20]], 95], [[a0, a2, a4, s], [[[a0, t], 17], [[a2, t], 19], [[a4, t], 21], [[s, a1], 18], [[s, a3], 20]], 95], [[a1, a2, a4, s], [[[a1, t], 18], [[a2, t], 19], [[a4, t], 21], [[s, a0], 17], [[s, a3], 20]], 95], [[a0, a1, a2, a4, s], [[[a0, t], 17], [[a1, t], 18], [[a2, t], 19], [[a4, t], 21], [[s, a3], 20]], 95], [[a3, a4, s], [[[a3, t], 20], [[a4, t], 21], [[s, a0], 17], [[s, a1], 18], [[s, a2], 19]], 95], [[a0, a3, a4, s], [[[a0, t], 17], [[a3, t], 20], [[a4, t], 21], [[s, a1], 18], [[s, a2], 19]], 95], [[a1, a3, a4, s], [[[a1, t], 18], [[a3, t], 20], [[a4, t], 21], [[s, a0], 17], [[s, a2], 19]], 95], [[a0, a1, a3, a4, s], [[[a0, t], 17], [[a1, t], 18], [[a3, t], 20], [[a4, t], 21], [[s, a2], 19]], 95], [[a2, a3, a4, s], [[[a2, t], 19], [[a3, t], 20], [[a4, t], 21], [[s, a0], 17], [[s, a1], 18]], 95], [[a0, a2, a3, a4, s], [[[a0, t], 17], [[a2, t], 19], [[a3, t], 20], [[a4, t], 21], [[s, a1], 18]], 95], [[a1, a2, a3, a4, s], [[[a1, t], 18], [[a2, t], 19], [[a3, t], 20], [[a4, t], 21], [[s, a0], 17]], 95], [[a0, a1, a2, a3, a4, s], [[[a0, t], 17], [[a1, t], 18], [[a2, t], 19], [[a3, t], 20], [[a4, t], 21]], 95]], 32, 32, every_subset_containing_s_and_excluding_t_is_enumerated_and_only_forward_edges_are_counted).
Reviewed step-by-step derivation
Max-flow min-cut theorem · qf-mod46-mfmc-complete-cut-enumeration · candidate
17. Every Source-Side Vertex Subset Defines One Directed Cut Audit
Use complete directed vertices=[s, a0, a1, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t]], source=s, sink=t, aligned nonnegative exact capacities=[18, 18, 19, 19]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (vertices, source, sink, cut_rows, cut_count, required_cut_count, certificate).
View answer and explanation
([a0, a1, s, t], s, t, [[[s], [[[s, a0], 18], [[s, a1], 19]], 37], [[a0, s], [[[a0, t], 18], [[s, a1], 19]], 37], [[a1, s], [[[a1, t], 19], [[s, a0], 18]], 37], [[a0, a1, s], [[[a0, t], 18], [[a1, t], 19]], 37]], 4, 4, every_subset_containing_s_and_excluding_t_is_enumerated_and_only_forward_edges_are_counted)
Enumerate every subset of the other vertices, with source included and sink excluded. For each cut list every outgoing edge and its capacity, then sum exactly. Retain actual and required counts; incoming edges are not charged to cut capacity. The exact result, retaining the requested order, is ([a0, a1, s, t], s, t, [[[s], [[[s, a0], 18], [[s, a1], 19]], 37], [[a0, s], [[[a0, t], 18], [[s, a1], 19]], 37], [[a1, s], [[[a1, t], 19], [[s, a0], 18]], 37], [[a0, a1, s], [[[a0, t], 18], [[a1, t], 19]], 37]], 4, 4, every_subset_containing_s_and_excluding_t_is_enumerated_and_only_forward_edges_are_counted).
Reviewed step-by-step derivation
Max-flow min-cut theorem · qf-mod46-mfmc-complete-cut-enumeration · candidate
18. Every Source-Side Vertex Subset Defines One Directed Cut Audit
Use complete directed vertices=[s, a0, a1, a2, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t], [s, a2], [a2, t]], source=s, sink=t, aligned nonnegative exact capacities=[19, 19, 20, 20, 21, 21]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (vertices, source, sink, cut_rows, cut_count, required_cut_count, certificate).
View answer and explanation
([a0, a1, a2, s, t], s, t, [[[s], [[[s, a0], 19], [[s, a1], 20], [[s, a2], 21]], 60], [[a0, s], [[[a0, t], 19], [[s, a1], 20], [[s, a2], 21]], 60], [[a1, s], [[[a1, t], 20], [[s, a0], 19], [[s, a2], 21]], 60], [[a0, a1, s], [[[a0, t], 19], [[a1, t], 20], [[s, a2], 21]], 60], [[a2, s], [[[a2, t], 21], [[s, a0], 19], [[s, a1], 20]], 60], [[a0, a2, s], [[[a0, t], 19], [[a2, t], 21], [[s, a1], 20]], 60], [[a1, a2, s], [[[a1, t], 20], [[a2, t], 21], [[s, a0], 19]], 60], [[a0, a1, a2, s], [[[a0, t], 19], [[a1, t], 20], [[a2, t], 21]], 60]], 8, 8, every_subset_containing_s_and_excluding_t_is_enumerated_and_only_forward_edges_are_counted)
Enumerate every subset of the other vertices, with source included and sink excluded. For each cut list every outgoing edge and its capacity, then sum exactly. Retain actual and required counts; incoming edges are not charged to cut capacity. The exact result, retaining the requested order, is ([a0, a1, a2, s, t], s, t, [[[s], [[[s, a0], 19], [[s, a1], 20], [[s, a2], 21]], 60], [[a0, s], [[[a0, t], 19], [[s, a1], 20], [[s, a2], 21]], 60], [[a1, s], [[[a1, t], 20], [[s, a0], 19], [[s, a2], 21]], 60], [[a0, a1, s], [[[a0, t], 19], [[a1, t], 20], [[s, a2], 21]], 60], [[a2, s], [[[a2, t], 21], [[s, a0], 19], [[s, a1], 20]], 60], [[a0, a2, s], [[[a0, t], 19], [[a2, t], 21], [[s, a1], 20]], 60], [[a1, a2, s], [[[a1, t], 20], [[a2, t], 21], [[s, a0], 19]], 60], [[a0, a1, a2, s], [[[a0, t], 19], [[a1, t], 20], [[a2, t], 21]], 60]], 8, 8, every_subset_containing_s_and_excluding_t_is_enumerated_and_only_forward_edges_are_counted).
Reviewed step-by-step derivation
Max-flow min-cut theorem · qf-mod46-mfmc-complete-cut-enumeration · candidate
19. Every Source-Side Vertex Subset Defines One Directed Cut Audit
Use complete directed vertices=[s, a0, a1, a2, a3, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t], [s, a2], [a2, t], [s, a3], [a3, t], [a0, s]], source=s, sink=t, aligned nonnegative exact capacities=[20, 20, 21, 21, 22, 22, 23, 23, 25]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (vertices, source, sink, cut_rows, cut_count, required_cut_count, certificate).
View answer and explanation
([a0, a1, a2, a3, s, t], s, t, [[[s], [[[s, a0], 20], [[s, a1], 21], [[s, a2], 22], [[s, a3], 23]], 86], [[a0, s], [[[a0, t], 20], [[s, a1], 21], [[s, a2], 22], [[s, a3], 23]], 86], [[a1, s], [[[a1, t], 21], [[s, a0], 20], [[s, a2], 22], [[s, a3], 23]], 86], [[a0, a1, s], [[[a0, t], 20], [[a1, t], 21], [[s, a2], 22], [[s, a3], 23]], 86], [[a2, s], [[[a2, t], 22], [[s, a0], 20], [[s, a1], 21], [[s, a3], 23]], 86], [[a0, a2, s], [[[a0, t], 20], [[a2, t], 22], [[s, a1], 21], [[s, a3], 23]], 86], [[a1, a2, s], [[[a1, t], 21], [[a2, t], 22], [[s, a0], 20], [[s, a3], 23]], 86], [[a0, a1, a2, s], [[[a0, t], 20], [[a1, t], 21], [[a2, t], 22], [[s, a3], 23]], 86], [[a3, s], [[[a3, t], 23], [[s, a0], 20], [[s, a1], 21], [[s, a2], 22]], 86], [[a0, a3, s], [[[a0, t], 20], [[a3, t], 23], [[s, a1], 21], [[s, a2], 22]], 86], [[a1, a3, s], [[[a1, t], 21], [[a3, t], 23], [[s, a0], 20], [[s, a2], 22]], 86], [[a0, a1, a3, s], [[[a0, t], 20], [[a1, t], 21], [[a3, t], 23], [[s, a2], 22]], 86], [[a2, a3, s], [[[a2, t], 22], [[a3, t], 23], [[s, a0], 20], [[s, a1], 21]], 86], [[a0, a2, a3, s], [[[a0, t], 20], [[a2, t], 22], [[a3, t], 23], [[s, a1], 21]], 86], [[a1, a2, a3, s], [[[a1, t], 21], [[a2, t], 22], [[a3, t], 23], [[s, a0], 20]], 86], [[a0, a1, a2, a3, s], [[[a0, t], 20], [[a1, t], 21], [[a2, t], 22], [[a3, t], 23]], 86]], 16, 16, every_subset_containing_s_and_excluding_t_is_enumerated_and_only_forward_edges_are_counted)
Enumerate every subset of the other vertices, with source included and sink excluded. For each cut list every outgoing edge and its capacity, then sum exactly. Retain actual and required counts; incoming edges are not charged to cut capacity. The exact result, retaining the requested order, is ([a0, a1, a2, a3, s, t], s, t, [[[s], [[[s, a0], 20], [[s, a1], 21], [[s, a2], 22], [[s, a3], 23]], 86], [[a0, s], [[[a0, t], 20], [[s, a1], 21], [[s, a2], 22], [[s, a3], 23]], 86], [[a1, s], [[[a1, t], 21], [[s, a0], 20], [[s, a2], 22], [[s, a3], 23]], 86], [[a0, a1, s], [[[a0, t], 20], [[a1, t], 21], [[s, a2], 22], [[s, a3], 23]], 86], [[a2, s], [[[a2, t], 22], [[s, a0], 20], [[s, a1], 21], [[s, a3], 23]], 86], [[a0, a2, s], [[[a0, t], 20], [[a2, t], 22], [[s, a1], 21], [[s, a3], 23]], 86], [[a1, a2, s], [[[a1, t], 21], [[a2, t], 22], [[s, a0], 20], [[s, a3], 23]], 86], [[a0, a1, a2, s], [[[a0, t], 20], [[a1, t], 21], [[a2, t], 22], [[s, a3], 23]], 86], [[a3, s], [[[a3, t], 23], [[s, a0], 20], [[s, a1], 21], [[s, a2], 22]], 86], [[a0, a3, s], [[[a0, t], 20], [[a3, t], 23], [[s, a1], 21], [[s, a2], 22]], 86], [[a1, a3, s], [[[a1, t], 21], [[a3, t], 23], [[s, a0], 20], [[s, a2], 22]], 86], [[a0, a1, a3, s], [[[a0, t], 20], [[a1, t], 21], [[a3, t], 23], [[s, a2], 22]], 86], [[a2, a3, s], [[[a2, t], 22], [[a3, t], 23], [[s, a0], 20], [[s, a1], 21]], 86], [[a0, a2, a3, s], [[[a0, t], 20], [[a2, t], 22], [[a3, t], 23], [[s, a1], 21]], 86], [[a1, a2, a3, s], [[[a1, t], 21], [[a2, t], 22], [[a3, t], 23], [[s, a0], 20]], 86], [[a0, a1, a2, a3, s], [[[a0, t], 20], [[a1, t], 21], [[a2, t], 22], [[a3, t], 23]], 86]], 16, 16, every_subset_containing_s_and_excluding_t_is_enumerated_and_only_forward_edges_are_counted).
Reviewed step-by-step derivation
Max-flow min-cut theorem · qf-mod46-mfmc-complete-cut-enumeration · candidate
20. Every Source-Side Vertex Subset Defines One Directed Cut Audit
Use complete directed vertices=[s, a0, a1, a2, a3, a4, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t], [s, a2], [a2, t], [s, a3], [a3, t], [s, a4], [a4, t]], source=s, sink=t, aligned nonnegative exact capacities=[21, 21, 22, 22, 23, 23, 24, 24, 25, 25]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (vertices, source, sink, cut_rows, cut_count, required_cut_count, certificate).
View answer and explanation
([a0, a1, a2, a3, a4, s, t], s, t, [[[s], [[[s, a0], 21], [[s, a1], 22], [[s, a2], 23], [[s, a3], 24], [[s, a4], 25]], 115], [[a0, s], [[[a0, t], 21], [[s, a1], 22], [[s, a2], 23], [[s, a3], 24], [[s, a4], 25]], 115], [[a1, s], [[[a1, t], 22], [[s, a0], 21], [[s, a2], 23], [[s, a3], 24], [[s, a4], 25]], 115], [[a0, a1, s], [[[a0, t], 21], [[a1, t], 22], [[s, a2], 23], [[s, a3], 24], [[s, a4], 25]], 115], [[a2, s], [[[a2, t], 23], [[s, a0], 21], [[s, a1], 22], [[s, a3], 24], [[s, a4], 25]], 115], [[a0, a2, s], [[[a0, t], 21], [[a2, t], 23], [[s, a1], 22], [[s, a3], 24], [[s, a4], 25]], 115], [[a1, a2, s], [[[a1, t], 22], [[a2, t], 23], [[s, a0], 21], [[s, a3], 24], [[s, a4], 25]], 115], [[a0, a1, a2, s], [[[a0, t], 21], [[a1, t], 22], [[a2, t], 23], [[s, a3], 24], [[s, a4], 25]], 115], [[a3, s], [[[a3, t], 24], [[s, a0], 21], [[s, a1], 22], [[s, a2], 23], [[s, a4], 25]], 115], [[a0, a3, s], [[[a0, t], 21], [[a3, t], 24], [[s, a1], 22], [[s, a2], 23], [[s, a4], 25]], 115], [[a1, a3, s], [[[a1, t], 22], [[a3, t], 24], [[s, a0], 21], [[s, a2], 23], [[s, a4], 25]], 115], [[a0, a1, a3, s], [[[a0, t], 21], [[a1, t], 22], [[a3, t], 24], [[s, a2], 23], [[s, a4], 25]], 115], [[a2, a3, s], [[[a2, t], 23], [[a3, t], 24], [[s, a0], 21], [[s, a1], 22], [[s, a4], 25]], 115], [[a0, a2, a3, s], [[[a0, t], 21], [[a2, t], 23], [[a3, t], 24], [[s, a1], 22], [[s, a4], 25]], 115], [[a1, a2, a3, s], [[[a1, t], 22], [[a2, t], 23], [[a3, t], 24], [[s, a0], 21], [[s, a4], 25]], 115], [[a0, a1, a2, a3, s], [[[a0, t], 21], [[a1, t], 22], [[a2, t], 23], [[a3, t], 24], [[s, a4], 25]], 115], [[a4, s], [[[a4, t], 25], [[s, a0], 21], [[s, a1], 22], [[s, a2], 23], [[s, a3], 24]], 115], [[a0, a4, s], [[[a0, t], 21], [[a4, t], 25], [[s, a1], 22], [[s, a2], 23], [[s, a3], 24]], 115], [[a1, a4, s], [[[a1, t], 22], [[a4, t], 25], [[s, a0], 21], [[s, a2], 23], [[s, a3], 24]], 115], [[a0, a1, a4, s], [[[a0, t], 21], [[a1, t], 22], [[a4, t], 25], [[s, a2], 23], [[s, a3], 24]], 115], [[a2, a4, s], [[[a2, t], 23], [[a4, t], 25], [[s, a0], 21], [[s, a1], 22], [[s, a3], 24]], 115], [[a0, a2, a4, s], [[[a0, t], 21], [[a2, t], 23], [[a4, t], 25], [[s, a1], 22], [[s, a3], 24]], 115], [[a1, a2, a4, s], [[[a1, t], 22], [[a2, t], 23], [[a4, t], 25], [[s, a0], 21], [[s, a3], 24]], 115], [[a0, a1, a2, a4, s], [[[a0, t], 21], [[a1, t], 22], [[a2, t], 23], [[a4, t], 25], [[s, a3], 24]], 115], [[a3, a4, s], [[[a3, t], 24], [[a4, t], 25], [[s, a0], 21], [[s, a1], 22], [[s, a2], 23]], 115], [[a0, a3, a4, s], [[[a0, t], 21], [[a3, t], 24], [[a4, t], 25], [[s, a1], 22], [[s, a2], 23]], 115], [[a1, a3, a4, s], [[[a1, t], 22], [[a3, t], 24], [[a4, t], 25], [[s, a0], 21], [[s, a2], 23]], 115], [[a0, a1, a3, a4, s], [[[a0, t], 21], [[a1, t], 22], [[a3, t], 24], [[a4, t], 25], [[s, a2], 23]], 115], [[a2, a3, a4, s], [[[a2, t], 23], [[a3, t], 24], [[a4, t], 25], [[s, a0], 21], [[s, a1], 22]], 115], [[a0, a2, a3, a4, s], [[[a0, t], 21], [[a2, t], 23], [[a3, t], 24], [[a4, t], 25], [[s, a1], 22]], 115], [[a1, a2, a3, a4, s], [[[a1, t], 22], [[a2, t], 23], [[a3, t], 24], [[a4, t], 25], [[s, a0], 21]], 115], [[a0, a1, a2, a3, a4, s], [[[a0, t], 21], [[a1, t], 22], [[a2, t], 23], [[a3, t], 24], [[a4, t], 25]], 115]], 32, 32, every_subset_containing_s_and_excluding_t_is_enumerated_and_only_forward_edges_are_counted)
Enumerate every subset of the other vertices, with source included and sink excluded. For each cut list every outgoing edge and its capacity, then sum exactly. Retain actual and required counts; incoming edges are not charged to cut capacity. The exact result, retaining the requested order, is ([a0, a1, a2, a3, a4, s, t], s, t, [[[s], [[[s, a0], 21], [[s, a1], 22], [[s, a2], 23], [[s, a3], 24], [[s, a4], 25]], 115], [[a0, s], [[[a0, t], 21], [[s, a1], 22], [[s, a2], 23], [[s, a3], 24], [[s, a4], 25]], 115], [[a1, s], [[[a1, t], 22], [[s, a0], 21], [[s, a2], 23], [[s, a3], 24], [[s, a4], 25]], 115], [[a0, a1, s], [[[a0, t], 21], [[a1, t], 22], [[s, a2], 23], [[s, a3], 24], [[s, a4], 25]], 115], [[a2, s], [[[a2, t], 23], [[s, a0], 21], [[s, a1], 22], [[s, a3], 24], [[s, a4], 25]], 115], [[a0, a2, s], [[[a0, t], 21], [[a2, t], 23], [[s, a1], 22], [[s, a3], 24], [[s, a4], 25]], 115], [[a1, a2, s], [[[a1, t], 22], [[a2, t], 23], [[s, a0], 21], [[s, a3], 24], [[s, a4], 25]], 115], [[a0, a1, a2, s], [[[a0, t], 21], [[a1, t], 22], [[a2, t], 23], [[s, a3], 24], [[s, a4], 25]], 115], [[a3, s], [[[a3, t], 24], [[s, a0], 21], [[s, a1], 22], [[s, a2], 23], [[s, a4], 25]], 115], [[a0, a3, s], [[[a0, t], 21], [[a3, t], 24], [[s, a1], 22], [[s, a2], 23], [[s, a4], 25]], 115], [[a1, a3, s], [[[a1, t], 22], [[a3, t], 24], [[s, a0], 21], [[s, a2], 23], [[s, a4], 25]], 115], [[a0, a1, a3, s], [[[a0, t], 21], [[a1, t], 22], [[a3, t], 24], [[s, a2], 23], [[s, a4], 25]], 115], [[a2, a3, s], [[[a2, t], 23], [[a3, t], 24], [[s, a0], 21], [[s, a1], 22], [[s, a4], 25]], 115], [[a0, a2, a3, s], [[[a0, t], 21], [[a2, t], 23], [[a3, t], 24], [[s, a1], 22], [[s, a4], 25]], 115], [[a1, a2, a3, s], [[[a1, t], 22], [[a2, t], 23], [[a3, t], 24], [[s, a0], 21], [[s, a4], 25]], 115], [[a0, a1, a2, a3, s], [[[a0, t], 21], [[a1, t], 22], [[a2, t], 23], [[a3, t], 24], [[s, a4], 25]], 115], [[a4, s], [[[a4, t], 25], [[s, a0], 21], [[s, a1], 22], [[s, a2], 23], [[s, a3], 24]], 115], [[a0, a4, s], [[[a0, t], 21], [[a4, t], 25], [[s, a1], 22], [[s, a2], 23], [[s, a3], 24]], 115], [[a1, a4, s], [[[a1, t], 22], [[a4, t], 25], [[s, a0], 21], [[s, a2], 23], [[s, a3], 24]], 115], [[a0, a1, a4, s], [[[a0, t], 21], [[a1, t], 22], [[a4, t], 25], [[s, a2], 23], [[s, a3], 24]], 115], [[a2, a4, s], [[[a2, t], 23], [[a4, t], 25], [[s, a0], 21], [[s, a1], 22], [[s, a3], 24]], 115], [[a0, a2, a4, s], [[[a0, t], 21], [[a2, t], 23], [[a4, t], 25], [[s, a1], 22], [[s, a3], 24]], 115], [[a1, a2, a4, s], [[[a1, t], 22], [[a2, t], 23], [[a4, t], 25], [[s, a0], 21], [[s, a3], 24]], 115], [[a0, a1, a2, a4, s], [[[a0, t], 21], [[a1, t], 22], [[a2, t], 23], [[a4, t], 25], [[s, a3], 24]], 115], [[a3, a4, s], [[[a3, t], 24], [[a4, t], 25], [[s, a0], 21], [[s, a1], 22], [[s, a2], 23]], 115], [[a0, a3, a4, s], [[[a0, t], 21], [[a3, t], 24], [[a4, t], 25], [[s, a1], 22], [[s, a2], 23]], 115], [[a1, a3, a4, s], [[[a1, t], 22], [[a3, t], 24], [[a4, t], 25], [[s, a0], 21], [[s, a2], 23]], 115], [[a0, a1, a3, a4, s], [[[a0, t], 21], [[a1, t], 22], [[a3, t], 24], [[a4, t], 25], [[s, a2], 23]], 115], [[a2, a3, a4, s], [[[a2, t], 23], [[a3, t], 24], [[a4, t], 25], [[s, a0], 21], [[s, a1], 22]], 115], [[a0, a2, a3, a4, s], [[[a0, t], 21], [[a2, t], 23], [[a3, t], 24], [[a4, t], 25], [[s, a1], 22]], 115], [[a1, a2, a3, a4, s], [[[a1, t], 22], [[a2, t], 23], [[a3, t], 24], [[a4, t], 25], [[s, a0], 21]], 115], [[a0, a1, a2, a3, a4, s], [[[a0, t], 21], [[a1, t], 22], [[a2, t], 23], [[a3, t], 24], [[a4, t], 25]], 115]], 32, 32, every_subset_containing_s_and_excluding_t_is_enumerated_and_only_forward_edges_are_counted).
Reviewed step-by-step derivation
Max-flow min-cut theorem · qf-mod46-mfmc-complete-cut-enumeration · candidate
21. Every Source-Side Vertex Subset Defines One Directed Cut Audit
Use complete directed vertices=[s, a0, a1, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t]], source=s, sink=t, aligned nonnegative exact capacities=[22, 22, 23, 23]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (vertices, source, sink, cut_rows, cut_count, required_cut_count, certificate).
View answer and explanation
([a0, a1, s, t], s, t, [[[s], [[[s, a0], 22], [[s, a1], 23]], 45], [[a0, s], [[[a0, t], 22], [[s, a1], 23]], 45], [[a1, s], [[[a1, t], 23], [[s, a0], 22]], 45], [[a0, a1, s], [[[a0, t], 22], [[a1, t], 23]], 45]], 4, 4, every_subset_containing_s_and_excluding_t_is_enumerated_and_only_forward_edges_are_counted)
Enumerate every subset of the other vertices, with source included and sink excluded. For each cut list every outgoing edge and its capacity, then sum exactly. Retain actual and required counts; incoming edges are not charged to cut capacity. The exact result, retaining the requested order, is ([a0, a1, s, t], s, t, [[[s], [[[s, a0], 22], [[s, a1], 23]], 45], [[a0, s], [[[a0, t], 22], [[s, a1], 23]], 45], [[a1, s], [[[a1, t], 23], [[s, a0], 22]], 45], [[a0, a1, s], [[[a0, t], 22], [[a1, t], 23]], 45]], 4, 4, every_subset_containing_s_and_excluding_t_is_enumerated_and_only_forward_edges_are_counted).
Reviewed step-by-step derivation
Max-flow min-cut theorem · qf-mod46-mfmc-complete-cut-enumeration · candidate
22. Every Source-Side Vertex Subset Defines One Directed Cut Audit
Use complete directed vertices=[s, a0, a1, a2, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t], [s, a2], [a2, t], [a0, s]], source=s, sink=t, aligned nonnegative exact capacities=[23, 23, 24, 24, 25, 25, 28]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (vertices, source, sink, cut_rows, cut_count, required_cut_count, certificate).
View answer and explanation
([a0, a1, a2, s, t], s, t, [[[s], [[[s, a0], 23], [[s, a1], 24], [[s, a2], 25]], 72], [[a0, s], [[[a0, t], 23], [[s, a1], 24], [[s, a2], 25]], 72], [[a1, s], [[[a1, t], 24], [[s, a0], 23], [[s, a2], 25]], 72], [[a0, a1, s], [[[a0, t], 23], [[a1, t], 24], [[s, a2], 25]], 72], [[a2, s], [[[a2, t], 25], [[s, a0], 23], [[s, a1], 24]], 72], [[a0, a2, s], [[[a0, t], 23], [[a2, t], 25], [[s, a1], 24]], 72], [[a1, a2, s], [[[a1, t], 24], [[a2, t], 25], [[s, a0], 23]], 72], [[a0, a1, a2, s], [[[a0, t], 23], [[a1, t], 24], [[a2, t], 25]], 72]], 8, 8, every_subset_containing_s_and_excluding_t_is_enumerated_and_only_forward_edges_are_counted)
Enumerate every subset of the other vertices, with source included and sink excluded. For each cut list every outgoing edge and its capacity, then sum exactly. Retain actual and required counts; incoming edges are not charged to cut capacity. The exact result, retaining the requested order, is ([a0, a1, a2, s, t], s, t, [[[s], [[[s, a0], 23], [[s, a1], 24], [[s, a2], 25]], 72], [[a0, s], [[[a0, t], 23], [[s, a1], 24], [[s, a2], 25]], 72], [[a1, s], [[[a1, t], 24], [[s, a0], 23], [[s, a2], 25]], 72], [[a0, a1, s], [[[a0, t], 23], [[a1, t], 24], [[s, a2], 25]], 72], [[a2, s], [[[a2, t], 25], [[s, a0], 23], [[s, a1], 24]], 72], [[a0, a2, s], [[[a0, t], 23], [[a2, t], 25], [[s, a1], 24]], 72], [[a1, a2, s], [[[a1, t], 24], [[a2, t], 25], [[s, a0], 23]], 72], [[a0, a1, a2, s], [[[a0, t], 23], [[a1, t], 24], [[a2, t], 25]], 72]], 8, 8, every_subset_containing_s_and_excluding_t_is_enumerated_and_only_forward_edges_are_counted).
Reviewed step-by-step derivation
Max-flow min-cut theorem · qf-mod46-mfmc-complete-cut-enumeration · candidate
23. Every Source-Side Vertex Subset Defines One Directed Cut Audit
Use complete directed vertices=[s, a0, a1, a2, a3, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t], [s, a2], [a2, t], [s, a3], [a3, t]], source=s, sink=t, aligned nonnegative exact capacities=[24, 24, 25, 25, 26, 26, 27, 27]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (vertices, source, sink, cut_rows, cut_count, required_cut_count, certificate).
View answer and explanation
([a0, a1, a2, a3, s, t], s, t, [[[s], [[[s, a0], 24], [[s, a1], 25], [[s, a2], 26], [[s, a3], 27]], 102], [[a0, s], [[[a0, t], 24], [[s, a1], 25], [[s, a2], 26], [[s, a3], 27]], 102], [[a1, s], [[[a1, t], 25], [[s, a0], 24], [[s, a2], 26], [[s, a3], 27]], 102], [[a0, a1, s], [[[a0, t], 24], [[a1, t], 25], [[s, a2], 26], [[s, a3], 27]], 102], [[a2, s], [[[a2, t], 26], [[s, a0], 24], [[s, a1], 25], [[s, a3], 27]], 102], [[a0, a2, s], [[[a0, t], 24], [[a2, t], 26], [[s, a1], 25], [[s, a3], 27]], 102], [[a1, a2, s], [[[a1, t], 25], [[a2, t], 26], [[s, a0], 24], [[s, a3], 27]], 102], [[a0, a1, a2, s], [[[a0, t], 24], [[a1, t], 25], [[a2, t], 26], [[s, a3], 27]], 102], [[a3, s], [[[a3, t], 27], [[s, a0], 24], [[s, a1], 25], [[s, a2], 26]], 102], [[a0, a3, s], [[[a0, t], 24], [[a3, t], 27], [[s, a1], 25], [[s, a2], 26]], 102], [[a1, a3, s], [[[a1, t], 25], [[a3, t], 27], [[s, a0], 24], [[s, a2], 26]], 102], [[a0, a1, a3, s], [[[a0, t], 24], [[a1, t], 25], [[a3, t], 27], [[s, a2], 26]], 102], [[a2, a3, s], [[[a2, t], 26], [[a3, t], 27], [[s, a0], 24], [[s, a1], 25]], 102], [[a0, a2, a3, s], [[[a0, t], 24], [[a2, t], 26], [[a3, t], 27], [[s, a1], 25]], 102], [[a1, a2, a3, s], [[[a1, t], 25], [[a2, t], 26], [[a3, t], 27], [[s, a0], 24]], 102], [[a0, a1, a2, a3, s], [[[a0, t], 24], [[a1, t], 25], [[a2, t], 26], [[a3, t], 27]], 102]], 16, 16, every_subset_containing_s_and_excluding_t_is_enumerated_and_only_forward_edges_are_counted)
Enumerate every subset of the other vertices, with source included and sink excluded. For each cut list every outgoing edge and its capacity, then sum exactly. Retain actual and required counts; incoming edges are not charged to cut capacity. The exact result, retaining the requested order, is ([a0, a1, a2, a3, s, t], s, t, [[[s], [[[s, a0], 24], [[s, a1], 25], [[s, a2], 26], [[s, a3], 27]], 102], [[a0, s], [[[a0, t], 24], [[s, a1], 25], [[s, a2], 26], [[s, a3], 27]], 102], [[a1, s], [[[a1, t], 25], [[s, a0], 24], [[s, a2], 26], [[s, a3], 27]], 102], [[a0, a1, s], [[[a0, t], 24], [[a1, t], 25], [[s, a2], 26], [[s, a3], 27]], 102], [[a2, s], [[[a2, t], 26], [[s, a0], 24], [[s, a1], 25], [[s, a3], 27]], 102], [[a0, a2, s], [[[a0, t], 24], [[a2, t], 26], [[s, a1], 25], [[s, a3], 27]], 102], [[a1, a2, s], [[[a1, t], 25], [[a2, t], 26], [[s, a0], 24], [[s, a3], 27]], 102], [[a0, a1, a2, s], [[[a0, t], 24], [[a1, t], 25], [[a2, t], 26], [[s, a3], 27]], 102], [[a3, s], [[[a3, t], 27], [[s, a0], 24], [[s, a1], 25], [[s, a2], 26]], 102], [[a0, a3, s], [[[a0, t], 24], [[a3, t], 27], [[s, a1], 25], [[s, a2], 26]], 102], [[a1, a3, s], [[[a1, t], 25], [[a3, t], 27], [[s, a0], 24], [[s, a2], 26]], 102], [[a0, a1, a3, s], [[[a0, t], 24], [[a1, t], 25], [[a3, t], 27], [[s, a2], 26]], 102], [[a2, a3, s], [[[a2, t], 26], [[a3, t], 27], [[s, a0], 24], [[s, a1], 25]], 102], [[a0, a2, a3, s], [[[a0, t], 24], [[a2, t], 26], [[a3, t], 27], [[s, a1], 25]], 102], [[a1, a2, a3, s], [[[a1, t], 25], [[a2, t], 26], [[a3, t], 27], [[s, a0], 24]], 102], [[a0, a1, a2, a3, s], [[[a0, t], 24], [[a1, t], 25], [[a2, t], 26], [[a3, t], 27]], 102]], 16, 16, every_subset_containing_s_and_excluding_t_is_enumerated_and_only_forward_edges_are_counted).
Reviewed step-by-step derivation
Max-flow min-cut theorem · qf-mod46-mfmc-complete-cut-enumeration · candidate
24. Every Source-Side Vertex Subset Defines One Directed Cut Audit
Use complete directed vertices=[s, a0, a1, a2, a3, a4, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t], [s, a2], [a2, t], [s, a3], [a3, t], [s, a4], [a4, t]], source=s, sink=t, aligned nonnegative exact capacities=[25, 25, 26, 26, 27, 27, 28, 28, 29, 29]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (vertices, source, sink, cut_rows, cut_count, required_cut_count, certificate).
View answer and explanation
([a0, a1, a2, a3, a4, s, t], s, t, [[[s], [[[s, a0], 25], [[s, a1], 26], [[s, a2], 27], [[s, a3], 28], [[s, a4], 29]], 135], [[a0, s], [[[a0, t], 25], [[s, a1], 26], [[s, a2], 27], [[s, a3], 28], [[s, a4], 29]], 135], [[a1, s], [[[a1, t], 26], [[s, a0], 25], [[s, a2], 27], [[s, a3], 28], [[s, a4], 29]], 135], [[a0, a1, s], [[[a0, t], 25], [[a1, t], 26], [[s, a2], 27], [[s, a3], 28], [[s, a4], 29]], 135], [[a2, s], [[[a2, t], 27], [[s, a0], 25], [[s, a1], 26], [[s, a3], 28], [[s, a4], 29]], 135], [[a0, a2, s], [[[a0, t], 25], [[a2, t], 27], [[s, a1], 26], [[s, a3], 28], [[s, a4], 29]], 135], [[a1, a2, s], [[[a1, t], 26], [[a2, t], 27], [[s, a0], 25], [[s, a3], 28], [[s, a4], 29]], 135], [[a0, a1, a2, s], [[[a0, t], 25], [[a1, t], 26], [[a2, t], 27], [[s, a3], 28], [[s, a4], 29]], 135], [[a3, s], [[[a3, t], 28], [[s, a0], 25], [[s, a1], 26], [[s, a2], 27], [[s, a4], 29]], 135], [[a0, a3, s], [[[a0, t], 25], [[a3, t], 28], [[s, a1], 26], [[s, a2], 27], [[s, a4], 29]], 135], [[a1, a3, s], [[[a1, t], 26], [[a3, t], 28], [[s, a0], 25], [[s, a2], 27], [[s, a4], 29]], 135], [[a0, a1, a3, s], [[[a0, t], 25], [[a1, t], 26], [[a3, t], 28], [[s, a2], 27], [[s, a4], 29]], 135], [[a2, a3, s], [[[a2, t], 27], [[a3, t], 28], [[s, a0], 25], [[s, a1], 26], [[s, a4], 29]], 135], [[a0, a2, a3, s], [[[a0, t], 25], [[a2, t], 27], [[a3, t], 28], [[s, a1], 26], [[s, a4], 29]], 135], [[a1, a2, a3, s], [[[a1, t], 26], [[a2, t], 27], [[a3, t], 28], [[s, a0], 25], [[s, a4], 29]], 135], [[a0, a1, a2, a3, s], [[[a0, t], 25], [[a1, t], 26], [[a2, t], 27], [[a3, t], 28], [[s, a4], 29]], 135], [[a4, s], [[[a4, t], 29], [[s, a0], 25], [[s, a1], 26], [[s, a2], 27], [[s, a3], 28]], 135], [[a0, a4, s], [[[a0, t], 25], [[a4, t], 29], [[s, a1], 26], [[s, a2], 27], [[s, a3], 28]], 135], [[a1, a4, s], [[[a1, t], 26], [[a4, t], 29], [[s, a0], 25], [[s, a2], 27], [[s, a3], 28]], 135], [[a0, a1, a4, s], [[[a0, t], 25], [[a1, t], 26], [[a4, t], 29], [[s, a2], 27], [[s, a3], 28]], 135], [[a2, a4, s], [[[a2, t], 27], [[a4, t], 29], [[s, a0], 25], [[s, a1], 26], [[s, a3], 28]], 135], [[a0, a2, a4, s], [[[a0, t], 25], [[a2, t], 27], [[a4, t], 29], [[s, a1], 26], [[s, a3], 28]], 135], [[a1, a2, a4, s], [[[a1, t], 26], [[a2, t], 27], [[a4, t], 29], [[s, a0], 25], [[s, a3], 28]], 135], [[a0, a1, a2, a4, s], [[[a0, t], 25], [[a1, t], 26], [[a2, t], 27], [[a4, t], 29], [[s, a3], 28]], 135], [[a3, a4, s], [[[a3, t], 28], [[a4, t], 29], [[s, a0], 25], [[s, a1], 26], [[s, a2], 27]], 135], [[a0, a3, a4, s], [[[a0, t], 25], [[a3, t], 28], [[a4, t], 29], [[s, a1], 26], [[s, a2], 27]], 135], [[a1, a3, a4, s], [[[a1, t], 26], [[a3, t], 28], [[a4, t], 29], [[s, a0], 25], [[s, a2], 27]], 135], [[a0, a1, a3, a4, s], [[[a0, t], 25], [[a1, t], 26], [[a3, t], 28], [[a4, t], 29], [[s, a2], 27]], 135], [[a2, a3, a4, s], [[[a2, t], 27], [[a3, t], 28], [[a4, t], 29], [[s, a0], 25], [[s, a1], 26]], 135], [[a0, a2, a3, a4, s], [[[a0, t], 25], [[a2, t], 27], [[a3, t], 28], [[a4, t], 29], [[s, a1], 26]], 135], [[a1, a2, a3, a4, s], [[[a1, t], 26], [[a2, t], 27], [[a3, t], 28], [[a4, t], 29], [[s, a0], 25]], 135], [[a0, a1, a2, a3, a4, s], [[[a0, t], 25], [[a1, t], 26], [[a2, t], 27], [[a3, t], 28], [[a4, t], 29]], 135]], 32, 32, every_subset_containing_s_and_excluding_t_is_enumerated_and_only_forward_edges_are_counted)
Enumerate every subset of the other vertices, with source included and sink excluded. For each cut list every outgoing edge and its capacity, then sum exactly. Retain actual and required counts; incoming edges are not charged to cut capacity. The exact result, retaining the requested order, is ([a0, a1, a2, a3, a4, s, t], s, t, [[[s], [[[s, a0], 25], [[s, a1], 26], [[s, a2], 27], [[s, a3], 28], [[s, a4], 29]], 135], [[a0, s], [[[a0, t], 25], [[s, a1], 26], [[s, a2], 27], [[s, a3], 28], [[s, a4], 29]], 135], [[a1, s], [[[a1, t], 26], [[s, a0], 25], [[s, a2], 27], [[s, a3], 28], [[s, a4], 29]], 135], [[a0, a1, s], [[[a0, t], 25], [[a1, t], 26], [[s, a2], 27], [[s, a3], 28], [[s, a4], 29]], 135], [[a2, s], [[[a2, t], 27], [[s, a0], 25], [[s, a1], 26], [[s, a3], 28], [[s, a4], 29]], 135], [[a0, a2, s], [[[a0, t], 25], [[a2, t], 27], [[s, a1], 26], [[s, a3], 28], [[s, a4], 29]], 135], [[a1, a2, s], [[[a1, t], 26], [[a2, t], 27], [[s, a0], 25], [[s, a3], 28], [[s, a4], 29]], 135], [[a0, a1, a2, s], [[[a0, t], 25], [[a1, t], 26], [[a2, t], 27], [[s, a3], 28], [[s, a4], 29]], 135], [[a3, s], [[[a3, t], 28], [[s, a0], 25], [[s, a1], 26], [[s, a2], 27], [[s, a4], 29]], 135], [[a0, a3, s], [[[a0, t], 25], [[a3, t], 28], [[s, a1], 26], [[s, a2], 27], [[s, a4], 29]], 135], [[a1, a3, s], [[[a1, t], 26], [[a3, t], 28], [[s, a0], 25], [[s, a2], 27], [[s, a4], 29]], 135], [[a0, a1, a3, s], [[[a0, t], 25], [[a1, t], 26], [[a3, t], 28], [[s, a2], 27], [[s, a4], 29]], 135], [[a2, a3, s], [[[a2, t], 27], [[a3, t], 28], [[s, a0], 25], [[s, a1], 26], [[s, a4], 29]], 135], [[a0, a2, a3, s], [[[a0, t], 25], [[a2, t], 27], [[a3, t], 28], [[s, a1], 26], [[s, a4], 29]], 135], [[a1, a2, a3, s], [[[a1, t], 26], [[a2, t], 27], [[a3, t], 28], [[s, a0], 25], [[s, a4], 29]], 135], [[a0, a1, a2, a3, s], [[[a0, t], 25], [[a1, t], 26], [[a2, t], 27], [[a3, t], 28], [[s, a4], 29]], 135], [[a4, s], [[[a4, t], 29], [[s, a0], 25], [[s, a1], 26], [[s, a2], 27], [[s, a3], 28]], 135], [[a0, a4, s], [[[a0, t], 25], [[a4, t], 29], [[s, a1], 26], [[s, a2], 27], [[s, a3], 28]], 135], [[a1, a4, s], [[[a1, t], 26], [[a4, t], 29], [[s, a0], 25], [[s, a2], 27], [[s, a3], 28]], 135], [[a0, a1, a4, s], [[[a0, t], 25], [[a1, t], 26], [[a4, t], 29], [[s, a2], 27], [[s, a3], 28]], 135], [[a2, a4, s], [[[a2, t], 27], [[a4, t], 29], [[s, a0], 25], [[s, a1], 26], [[s, a3], 28]], 135], [[a0, a2, a4, s], [[[a0, t], 25], [[a2, t], 27], [[a4, t], 29], [[s, a1], 26], [[s, a3], 28]], 135], [[a1, a2, a4, s], [[[a1, t], 26], [[a2, t], 27], [[a4, t], 29], [[s, a0], 25], [[s, a3], 28]], 135], [[a0, a1, a2, a4, s], [[[a0, t], 25], [[a1, t], 26], [[a2, t], 27], [[a4, t], 29], [[s, a3], 28]], 135], [[a3, a4, s], [[[a3, t], 28], [[a4, t], 29], [[s, a0], 25], [[s, a1], 26], [[s, a2], 27]], 135], [[a0, a3, a4, s], [[[a0, t], 25], [[a3, t], 28], [[a4, t], 29], [[s, a1], 26], [[s, a2], 27]], 135], [[a1, a3, a4, s], [[[a1, t], 26], [[a3, t], 28], [[a4, t], 29], [[s, a0], 25], [[s, a2], 27]], 135], [[a0, a1, a3, a4, s], [[[a0, t], 25], [[a1, t], 26], [[a3, t], 28], [[a4, t], 29], [[s, a2], 27]], 135], [[a2, a3, a4, s], [[[a2, t], 27], [[a3, t], 28], [[a4, t], 29], [[s, a0], 25], [[s, a1], 26]], 135], [[a0, a2, a3, a4, s], [[[a0, t], 25], [[a2, t], 27], [[a3, t], 28], [[a4, t], 29], [[s, a1], 26]], 135], [[a1, a2, a3, a4, s], [[[a1, t], 26], [[a2, t], 27], [[a3, t], 28], [[a4, t], 29], [[s, a0], 25]], 135], [[a0, a1, a2, a3, a4, s], [[[a0, t], 25], [[a1, t], 26], [[a2, t], 27], [[a3, t], 28], [[a4, t], 29]], 135]], 32, 32, every_subset_containing_s_and_excluding_t_is_enumerated_and_only_forward_edges_are_counted).
Reviewed step-by-step derivation
Max-flow min-cut theorem · qf-mod46-mfmc-complete-cut-enumeration · candidate
25. Every Source-Side Vertex Subset Defines One Directed Cut Audit
Use complete directed vertices=[s, a0, a1, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t], [a0, s]], source=s, sink=t, aligned nonnegative exact capacities=[26, 26, 27, 27, 31]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (vertices, source, sink, cut_rows, cut_count, required_cut_count, certificate).
View answer and explanation
([a0, a1, s, t], s, t, [[[s], [[[s, a0], 26], [[s, a1], 27]], 53], [[a0, s], [[[a0, t], 26], [[s, a1], 27]], 53], [[a1, s], [[[a1, t], 27], [[s, a0], 26]], 53], [[a0, a1, s], [[[a0, t], 26], [[a1, t], 27]], 53]], 4, 4, every_subset_containing_s_and_excluding_t_is_enumerated_and_only_forward_edges_are_counted)
Enumerate every subset of the other vertices, with source included and sink excluded. For each cut list every outgoing edge and its capacity, then sum exactly. Retain actual and required counts; incoming edges are not charged to cut capacity. The exact result, retaining the requested order, is ([a0, a1, s, t], s, t, [[[s], [[[s, a0], 26], [[s, a1], 27]], 53], [[a0, s], [[[a0, t], 26], [[s, a1], 27]], 53], [[a1, s], [[[a1, t], 27], [[s, a0], 26]], 53], [[a0, a1, s], [[[a0, t], 26], [[a1, t], 27]], 53]], 4, 4, every_subset_containing_s_and_excluding_t_is_enumerated_and_only_forward_edges_are_counted).
Reviewed step-by-step derivation
Max-flow min-cut theorem · qf-mod46-mfmc-complete-cut-enumeration · candidate
26. A Feasible Flow Is Bounded above by Every Cut, Not Automatically Maximum
Use complete directed vertices=[s, a0, a1, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t], [a0, s]], source=s, sink=t, aligned nonnegative exact capacities=[2, 2, 3, 3, 7], candidate flows=[1, 1, 2, 2, 0]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (source_value, sink_value, flow_cut_rows, all_cut_bounds_hold, automatic_maximum_claim, certificate).
Cut capacity counts outgoing edges only; net flow subtracts incoming flow. No automatic_maximum_claim does not classify this particular flow as necessarily nonmaximum.
View answer and explanation
(3, 3, [[[s], 3, 0, 3, 5, Yes, Yes], [[a0, s], 3, 0, 3, 5, Yes, Yes], [[a1, s], 3, 0, 3, 5, Yes, Yes], [[a0, a1, s], 3, 0, 3, 5, Yes, Yes]], Yes, No, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum)
Require nonnegative capacity-bounded flows and conservation at every internal vertex. For every cut compute outgoing flow, incoming flow, their difference, capacity, net/source-value equality and the upper bound. The fixed No withholds automatic maximality; it does not assert that this particular feasible flow is nonmaximum. The exact result, retaining the requested order, is (3, 3, [[[s], 3, 0, 3, 5, Yes, Yes], [[a0, s], 3, 0, 3, 5, Yes, Yes], [[a1, s], 3, 0, 3, 5, Yes, Yes], [[a0, a1, s], 3, 0, 3, 5, Yes, Yes]], Yes, No, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum).
Reviewed step-by-step derivation
Max-flow min-cut theorem · qf-mod46-mfmc-flow-cut-upper-bound · candidate
27. A Feasible Flow Is Bounded above by Every Cut, Not Automatically Maximum
Use complete directed vertices=[s, a0, a1, a2, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t], [s, a2], [a2, t]], source=s, sink=t, aligned nonnegative exact capacities=[3, 3, 4, 4, 5, 5], candidate flows=[2, 2, 3, 3, 4, 4]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (source_value, sink_value, flow_cut_rows, all_cut_bounds_hold, automatic_maximum_claim, certificate).
Cut capacity counts outgoing edges only; net flow subtracts incoming flow. No automatic_maximum_claim does not classify this particular flow as necessarily nonmaximum.
View answer and explanation
(9, 9, [[[s], 9, 0, 9, 12, Yes, Yes], [[a0, s], 9, 0, 9, 12, Yes, Yes], [[a1, s], 9, 0, 9, 12, Yes, Yes], [[a0, a1, s], 9, 0, 9, 12, Yes, Yes], [[a2, s], 9, 0, 9, 12, Yes, Yes], [[a0, a2, s], 9, 0, 9, 12, Yes, Yes], [[a1, a2, s], 9, 0, 9, 12, Yes, Yes], [[a0, a1, a2, s], 9, 0, 9, 12, Yes, Yes]], Yes, No, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum)
Require nonnegative capacity-bounded flows and conservation at every internal vertex. For every cut compute outgoing flow, incoming flow, their difference, capacity, net/source-value equality and the upper bound. The fixed No withholds automatic maximality; it does not assert that this particular feasible flow is nonmaximum. The exact result, retaining the requested order, is (9, 9, [[[s], 9, 0, 9, 12, Yes, Yes], [[a0, s], 9, 0, 9, 12, Yes, Yes], [[a1, s], 9, 0, 9, 12, Yes, Yes], [[a0, a1, s], 9, 0, 9, 12, Yes, Yes], [[a2, s], 9, 0, 9, 12, Yes, Yes], [[a0, a2, s], 9, 0, 9, 12, Yes, Yes], [[a1, a2, s], 9, 0, 9, 12, Yes, Yes], [[a0, a1, a2, s], 9, 0, 9, 12, Yes, Yes]], Yes, No, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum).
Reviewed step-by-step derivation
Max-flow min-cut theorem · qf-mod46-mfmc-flow-cut-upper-bound · candidate
28. A Feasible Flow Is Bounded above by Every Cut, Not Automatically Maximum
Use complete directed vertices=[s, a0, a1, a2, a3, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t], [s, a2], [a2, t], [s, a3], [a3, t]], source=s, sink=t, aligned nonnegative exact capacities=[4, 4, 5, 5, 6, 6, 7, 7], candidate flows=[3, 3, 4, 4, 5, 5, 6, 6]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (source_value, sink_value, flow_cut_rows, all_cut_bounds_hold, automatic_maximum_claim, certificate).
Cut capacity counts outgoing edges only; net flow subtracts incoming flow. No automatic_maximum_claim does not classify this particular flow as necessarily nonmaximum.
View answer and explanation
(18, 18, [[[s], 18, 0, 18, 22, Yes, Yes], [[a0, s], 18, 0, 18, 22, Yes, Yes], [[a1, s], 18, 0, 18, 22, Yes, Yes], [[a0, a1, s], 18, 0, 18, 22, Yes, Yes], [[a2, s], 18, 0, 18, 22, Yes, Yes], [[a0, a2, s], 18, 0, 18, 22, Yes, Yes], [[a1, a2, s], 18, 0, 18, 22, Yes, Yes], [[a0, a1, a2, s], 18, 0, 18, 22, Yes, Yes], [[a3, s], 18, 0, 18, 22, Yes, Yes], [[a0, a3, s], 18, 0, 18, 22, Yes, Yes], [[a1, a3, s], 18, 0, 18, 22, Yes, Yes], [[a0, a1, a3, s], 18, 0, 18, 22, Yes, Yes], [[a2, a3, s], 18, 0, 18, 22, Yes, Yes], [[a0, a2, a3, s], 18, 0, 18, 22, Yes, Yes], [[a1, a2, a3, s], 18, 0, 18, 22, Yes, Yes], [[a0, a1, a2, a3, s], 18, 0, 18, 22, Yes, Yes]], Yes, No, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum)
Require nonnegative capacity-bounded flows and conservation at every internal vertex. For every cut compute outgoing flow, incoming flow, their difference, capacity, net/source-value equality and the upper bound. The fixed No withholds automatic maximality; it does not assert that this particular feasible flow is nonmaximum. The exact result, retaining the requested order, is (18, 18, [[[s], 18, 0, 18, 22, Yes, Yes], [[a0, s], 18, 0, 18, 22, Yes, Yes], [[a1, s], 18, 0, 18, 22, Yes, Yes], [[a0, a1, s], 18, 0, 18, 22, Yes, Yes], [[a2, s], 18, 0, 18, 22, Yes, Yes], [[a0, a2, s], 18, 0, 18, 22, Yes, Yes], [[a1, a2, s], 18, 0, 18, 22, Yes, Yes], [[a0, a1, a2, s], 18, 0, 18, 22, Yes, Yes], [[a3, s], 18, 0, 18, 22, Yes, Yes], [[a0, a3, s], 18, 0, 18, 22, Yes, Yes], [[a1, a3, s], 18, 0, 18, 22, Yes, Yes], [[a0, a1, a3, s], 18, 0, 18, 22, Yes, Yes], [[a2, a3, s], 18, 0, 18, 22, Yes, Yes], [[a0, a2, a3, s], 18, 0, 18, 22, Yes, Yes], [[a1, a2, a3, s], 18, 0, 18, 22, Yes, Yes], [[a0, a1, a2, a3, s], 18, 0, 18, 22, Yes, Yes]], Yes, No, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum).
Reviewed step-by-step derivation
Max-flow min-cut theorem · qf-mod46-mfmc-flow-cut-upper-bound · candidate
29. A Feasible Flow Is Bounded above by Every Cut, Not Automatically Maximum
Use complete directed vertices=[s, a0, a1, a2, a3, a4, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t], [s, a2], [a2, t], [s, a3], [a3, t], [s, a4], [a4, t], [a0, s]], source=s, sink=t, aligned nonnegative exact capacities=[5, 5, 6, 6, 7, 7, 8, 8, 9, 9, 10], candidate flows=[4, 4, 5, 5, 6, 6, 7, 7, 8, 8, 0]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (source_value, sink_value, flow_cut_rows, all_cut_bounds_hold, automatic_maximum_claim, certificate).
Cut capacity counts outgoing edges only; net flow subtracts incoming flow. No automatic_maximum_claim does not classify this particular flow as necessarily nonmaximum.
View answer and explanation
(30, 30, [[[s], 30, 0, 30, 35, Yes, Yes], [[a0, s], 30, 0, 30, 35, Yes, Yes], [[a1, s], 30, 0, 30, 35, Yes, Yes], [[a0, a1, s], 30, 0, 30, 35, Yes, Yes], [[a2, s], 30, 0, 30, 35, Yes, Yes], [[a0, a2, s], 30, 0, 30, 35, Yes, Yes], [[a1, a2, s], 30, 0, 30, 35, Yes, Yes], [[a0, a1, a2, s], 30, 0, 30, 35, Yes, Yes], [[a3, s], 30, 0, 30, 35, Yes, Yes], [[a0, a3, s], 30, 0, 30, 35, Yes, Yes], [[a1, a3, s], 30, 0, 30, 35, Yes, Yes], [[a0, a1, a3, s], 30, 0, 30, 35, Yes, Yes], [[a2, a3, s], 30, 0, 30, 35, Yes, Yes], [[a0, a2, a3, s], 30, 0, 30, 35, Yes, Yes], [[a1, a2, a3, s], 30, 0, 30, 35, Yes, Yes], [[a0, a1, a2, a3, s], 30, 0, 30, 35, Yes, Yes], [[a4, s], 30, 0, 30, 35, Yes, Yes], [[a0, a4, s], 30, 0, 30, 35, Yes, Yes], [[a1, a4, s], 30, 0, 30, 35, Yes, Yes], [[a0, a1, a4, s], 30, 0, 30, 35, Yes, Yes], [[a2, a4, s], 30, 0, 30, 35, Yes, Yes], [[a0, a2, a4, s], 30, 0, 30, 35, Yes, Yes], [[a1, a2, a4, s], 30, 0, 30, 35, Yes, Yes], [[a0, a1, a2, a4, s], 30, 0, 30, 35, Yes, Yes], [[a3, a4, s], 30, 0, 30, 35, Yes, Yes], [[a0, a3, a4, s], 30, 0, 30, 35, Yes, Yes], [[a1, a3, a4, s], 30, 0, 30, 35, Yes, Yes], [[a0, a1, a3, a4, s], 30, 0, 30, 35, Yes, Yes], [[a2, a3, a4, s], 30, 0, 30, 35, Yes, Yes], [[a0, a2, a3, a4, s], 30, 0, 30, 35, Yes, Yes], [[a1, a2, a3, a4, s], 30, 0, 30, 35, Yes, Yes], [[a0, a1, a2, a3, a4, s], 30, 0, 30, 35, Yes, Yes]], Yes, No, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum)
Require nonnegative capacity-bounded flows and conservation at every internal vertex. For every cut compute outgoing flow, incoming flow, their difference, capacity, net/source-value equality and the upper bound. The fixed No withholds automatic maximality; it does not assert that this particular feasible flow is nonmaximum. The exact result, retaining the requested order, is (30, 30, [[[s], 30, 0, 30, 35, Yes, Yes], [[a0, s], 30, 0, 30, 35, Yes, Yes], [[a1, s], 30, 0, 30, 35, Yes, Yes], [[a0, a1, s], 30, 0, 30, 35, Yes, Yes], [[a2, s], 30, 0, 30, 35, Yes, Yes], [[a0, a2, s], 30, 0, 30, 35, Yes, Yes], [[a1, a2, s], 30, 0, 30, 35, Yes, Yes], [[a0, a1, a2, s], 30, 0, 30, 35, Yes, Yes], [[a3, s], 30, 0, 30, 35, Yes, Yes], [[a0, a3, s], 30, 0, 30, 35, Yes, Yes], [[a1, a3, s], 30, 0, 30, 35, Yes, Yes], [[a0, a1, a3, s], 30, 0, 30, 35, Yes, Yes], [[a2, a3, s], 30, 0, 30, 35, Yes, Yes], [[a0, a2, a3, s], 30, 0, 30, 35, Yes, Yes], [[a1, a2, a3, s], 30, 0, 30, 35, Yes, Yes], [[a0, a1, a2, a3, s], 30, 0, 30, 35, Yes, Yes], [[a4, s], 30, 0, 30, 35, Yes, Yes], [[a0, a4, s], 30, 0, 30, 35, Yes, Yes], [[a1, a4, s], 30, 0, 30, 35, Yes, Yes], [[a0, a1, a4, s], 30, 0, 30, 35, Yes, Yes], [[a2, a4, s], 30, 0, 30, 35, Yes, Yes], [[a0, a2, a4, s], 30, 0, 30, 35, Yes, Yes], [[a1, a2, a4, s], 30, 0, 30, 35, Yes, Yes], [[a0, a1, a2, a4, s], 30, 0, 30, 35, Yes, Yes], [[a3, a4, s], 30, 0, 30, 35, Yes, Yes], [[a0, a3, a4, s], 30, 0, 30, 35, Yes, Yes], [[a1, a3, a4, s], 30, 0, 30, 35, Yes, Yes], [[a0, a1, a3, a4, s], 30, 0, 30, 35, Yes, Yes], [[a2, a3, a4, s], 30, 0, 30, 35, Yes, Yes], [[a0, a2, a3, a4, s], 30, 0, 30, 35, Yes, Yes], [[a1, a2, a3, a4, s], 30, 0, 30, 35, Yes, Yes], [[a0, a1, a2, a3, a4, s], 30, 0, 30, 35, Yes, Yes]], Yes, No, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum).
Reviewed step-by-step derivation
Max-flow min-cut theorem · qf-mod46-mfmc-flow-cut-upper-bound · candidate
30. A Feasible Flow Is Bounded above by Every Cut, Not Automatically Maximum
Use complete directed vertices=[s, a0, a1, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t]], source=s, sink=t, aligned nonnegative exact capacities=[6, 6, 7, 7], candidate flows=[5, 5, 6, 6]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (source_value, sink_value, flow_cut_rows, all_cut_bounds_hold, automatic_maximum_claim, certificate).
Cut capacity counts outgoing edges only; net flow subtracts incoming flow. No automatic_maximum_claim does not classify this particular flow as necessarily nonmaximum.
View answer and explanation
(11, 11, [[[s], 11, 0, 11, 13, Yes, Yes], [[a0, s], 11, 0, 11, 13, Yes, Yes], [[a1, s], 11, 0, 11, 13, Yes, Yes], [[a0, a1, s], 11, 0, 11, 13, Yes, Yes]], Yes, No, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum)
Require nonnegative capacity-bounded flows and conservation at every internal vertex. For every cut compute outgoing flow, incoming flow, their difference, capacity, net/source-value equality and the upper bound. The fixed No withholds automatic maximality; it does not assert that this particular feasible flow is nonmaximum. The exact result, retaining the requested order, is (11, 11, [[[s], 11, 0, 11, 13, Yes, Yes], [[a0, s], 11, 0, 11, 13, Yes, Yes], [[a1, s], 11, 0, 11, 13, Yes, Yes], [[a0, a1, s], 11, 0, 11, 13, Yes, Yes]], Yes, No, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum).
Reviewed step-by-step derivation
Max-flow min-cut theorem · qf-mod46-mfmc-flow-cut-upper-bound · candidate
31. A Feasible Flow Is Bounded above by Every Cut, Not Automatically Maximum
Use complete directed vertices=[s, a0, a1, a2, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t], [s, a2], [a2, t]], source=s, sink=t, aligned nonnegative exact capacities=[7, 7, 8, 8, 9, 9], candidate flows=[6, 6, 7, 7, 8, 8]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (source_value, sink_value, flow_cut_rows, all_cut_bounds_hold, automatic_maximum_claim, certificate).
Cut capacity counts outgoing edges only; net flow subtracts incoming flow. No automatic_maximum_claim does not classify this particular flow as necessarily nonmaximum.
View answer and explanation
(21, 21, [[[s], 21, 0, 21, 24, Yes, Yes], [[a0, s], 21, 0, 21, 24, Yes, Yes], [[a1, s], 21, 0, 21, 24, Yes, Yes], [[a0, a1, s], 21, 0, 21, 24, Yes, Yes], [[a2, s], 21, 0, 21, 24, Yes, Yes], [[a0, a2, s], 21, 0, 21, 24, Yes, Yes], [[a1, a2, s], 21, 0, 21, 24, Yes, Yes], [[a0, a1, a2, s], 21, 0, 21, 24, Yes, Yes]], Yes, No, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum)
Require nonnegative capacity-bounded flows and conservation at every internal vertex. For every cut compute outgoing flow, incoming flow, their difference, capacity, net/source-value equality and the upper bound. The fixed No withholds automatic maximality; it does not assert that this particular feasible flow is nonmaximum. The exact result, retaining the requested order, is (21, 21, [[[s], 21, 0, 21, 24, Yes, Yes], [[a0, s], 21, 0, 21, 24, Yes, Yes], [[a1, s], 21, 0, 21, 24, Yes, Yes], [[a0, a1, s], 21, 0, 21, 24, Yes, Yes], [[a2, s], 21, 0, 21, 24, Yes, Yes], [[a0, a2, s], 21, 0, 21, 24, Yes, Yes], [[a1, a2, s], 21, 0, 21, 24, Yes, Yes], [[a0, a1, a2, s], 21, 0, 21, 24, Yes, Yes]], Yes, No, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum).
Reviewed step-by-step derivation
Max-flow min-cut theorem · qf-mod46-mfmc-flow-cut-upper-bound · candidate
32. A Feasible Flow Is Bounded above by Every Cut, Not Automatically Maximum
Use complete directed vertices=[s, a0, a1, a2, a3, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t], [s, a2], [a2, t], [s, a3], [a3, t], [a0, s]], source=s, sink=t, aligned nonnegative exact capacities=[8, 8, 9, 9, 10, 10, 11, 11, 13], candidate flows=[7, 7, 8, 8, 9, 9, 10, 10, 0]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (source_value, sink_value, flow_cut_rows, all_cut_bounds_hold, automatic_maximum_claim, certificate).
Cut capacity counts outgoing edges only; net flow subtracts incoming flow. No automatic_maximum_claim does not classify this particular flow as necessarily nonmaximum.
View answer and explanation
(34, 34, [[[s], 34, 0, 34, 38, Yes, Yes], [[a0, s], 34, 0, 34, 38, Yes, Yes], [[a1, s], 34, 0, 34, 38, Yes, Yes], [[a0, a1, s], 34, 0, 34, 38, Yes, Yes], [[a2, s], 34, 0, 34, 38, Yes, Yes], [[a0, a2, s], 34, 0, 34, 38, Yes, Yes], [[a1, a2, s], 34, 0, 34, 38, Yes, Yes], [[a0, a1, a2, s], 34, 0, 34, 38, Yes, Yes], [[a3, s], 34, 0, 34, 38, Yes, Yes], [[a0, a3, s], 34, 0, 34, 38, Yes, Yes], [[a1, a3, s], 34, 0, 34, 38, Yes, Yes], [[a0, a1, a3, s], 34, 0, 34, 38, Yes, Yes], [[a2, a3, s], 34, 0, 34, 38, Yes, Yes], [[a0, a2, a3, s], 34, 0, 34, 38, Yes, Yes], [[a1, a2, a3, s], 34, 0, 34, 38, Yes, Yes], [[a0, a1, a2, a3, s], 34, 0, 34, 38, Yes, Yes]], Yes, No, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum)
Require nonnegative capacity-bounded flows and conservation at every internal vertex. For every cut compute outgoing flow, incoming flow, their difference, capacity, net/source-value equality and the upper bound. The fixed No withholds automatic maximality; it does not assert that this particular feasible flow is nonmaximum. The exact result, retaining the requested order, is (34, 34, [[[s], 34, 0, 34, 38, Yes, Yes], [[a0, s], 34, 0, 34, 38, Yes, Yes], [[a1, s], 34, 0, 34, 38, Yes, Yes], [[a0, a1, s], 34, 0, 34, 38, Yes, Yes], [[a2, s], 34, 0, 34, 38, Yes, Yes], [[a0, a2, s], 34, 0, 34, 38, Yes, Yes], [[a1, a2, s], 34, 0, 34, 38, Yes, Yes], [[a0, a1, a2, s], 34, 0, 34, 38, Yes, Yes], [[a3, s], 34, 0, 34, 38, Yes, Yes], [[a0, a3, s], 34, 0, 34, 38, Yes, Yes], [[a1, a3, s], 34, 0, 34, 38, Yes, Yes], [[a0, a1, a3, s], 34, 0, 34, 38, Yes, Yes], [[a2, a3, s], 34, 0, 34, 38, Yes, Yes], [[a0, a2, a3, s], 34, 0, 34, 38, Yes, Yes], [[a1, a2, a3, s], 34, 0, 34, 38, Yes, Yes], [[a0, a1, a2, a3, s], 34, 0, 34, 38, Yes, Yes]], Yes, No, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum).
Reviewed step-by-step derivation
Max-flow min-cut theorem · qf-mod46-mfmc-flow-cut-upper-bound · candidate
33. A Feasible Flow Is Bounded above by Every Cut, Not Automatically Maximum
Use complete directed vertices=[s, a0, a1, a2, a3, a4, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t], [s, a2], [a2, t], [s, a3], [a3, t], [s, a4], [a4, t]], source=s, sink=t, aligned nonnegative exact capacities=[9, 9, 10, 10, 11, 11, 12, 12, 13, 13], candidate flows=[8, 8, 9, 9, 10, 10, 11, 11, 12, 12]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (source_value, sink_value, flow_cut_rows, all_cut_bounds_hold, automatic_maximum_claim, certificate).
Cut capacity counts outgoing edges only; net flow subtracts incoming flow. No automatic_maximum_claim does not classify this particular flow as necessarily nonmaximum.
View answer and explanation
(50, 50, [[[s], 50, 0, 50, 55, Yes, Yes], [[a0, s], 50, 0, 50, 55, Yes, Yes], [[a1, s], 50, 0, 50, 55, Yes, Yes], [[a0, a1, s], 50, 0, 50, 55, Yes, Yes], [[a2, s], 50, 0, 50, 55, Yes, Yes], [[a0, a2, s], 50, 0, 50, 55, Yes, Yes], [[a1, a2, s], 50, 0, 50, 55, Yes, Yes], [[a0, a1, a2, s], 50, 0, 50, 55, Yes, Yes], [[a3, s], 50, 0, 50, 55, Yes, Yes], [[a0, a3, s], 50, 0, 50, 55, Yes, Yes], [[a1, a3, s], 50, 0, 50, 55, Yes, Yes], [[a0, a1, a3, s], 50, 0, 50, 55, Yes, Yes], [[a2, a3, s], 50, 0, 50, 55, Yes, Yes], [[a0, a2, a3, s], 50, 0, 50, 55, Yes, Yes], [[a1, a2, a3, s], 50, 0, 50, 55, Yes, Yes], [[a0, a1, a2, a3, s], 50, 0, 50, 55, Yes, Yes], [[a4, s], 50, 0, 50, 55, Yes, Yes], [[a0, a4, s], 50, 0, 50, 55, Yes, Yes], [[a1, a4, s], 50, 0, 50, 55, Yes, Yes], [[a0, a1, a4, s], 50, 0, 50, 55, Yes, Yes], [[a2, a4, s], 50, 0, 50, 55, Yes, Yes], [[a0, a2, a4, s], 50, 0, 50, 55, Yes, Yes], [[a1, a2, a4, s], 50, 0, 50, 55, Yes, Yes], [[a0, a1, a2, a4, s], 50, 0, 50, 55, Yes, Yes], [[a3, a4, s], 50, 0, 50, 55, Yes, Yes], [[a0, a3, a4, s], 50, 0, 50, 55, Yes, Yes], [[a1, a3, a4, s], 50, 0, 50, 55, Yes, Yes], [[a0, a1, a3, a4, s], 50, 0, 50, 55, Yes, Yes], [[a2, a3, a4, s], 50, 0, 50, 55, Yes, Yes], [[a0, a2, a3, a4, s], 50, 0, 50, 55, Yes, Yes], [[a1, a2, a3, a4, s], 50, 0, 50, 55, Yes, Yes], [[a0, a1, a2, a3, a4, s], 50, 0, 50, 55, Yes, Yes]], Yes, No, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum)
Require nonnegative capacity-bounded flows and conservation at every internal vertex. For every cut compute outgoing flow, incoming flow, their difference, capacity, net/source-value equality and the upper bound. The fixed No withholds automatic maximality; it does not assert that this particular feasible flow is nonmaximum. The exact result, retaining the requested order, is (50, 50, [[[s], 50, 0, 50, 55, Yes, Yes], [[a0, s], 50, 0, 50, 55, Yes, Yes], [[a1, s], 50, 0, 50, 55, Yes, Yes], [[a0, a1, s], 50, 0, 50, 55, Yes, Yes], [[a2, s], 50, 0, 50, 55, Yes, Yes], [[a0, a2, s], 50, 0, 50, 55, Yes, Yes], [[a1, a2, s], 50, 0, 50, 55, Yes, Yes], [[a0, a1, a2, s], 50, 0, 50, 55, Yes, Yes], [[a3, s], 50, 0, 50, 55, Yes, Yes], [[a0, a3, s], 50, 0, 50, 55, Yes, Yes], [[a1, a3, s], 50, 0, 50, 55, Yes, Yes], [[a0, a1, a3, s], 50, 0, 50, 55, Yes, Yes], [[a2, a3, s], 50, 0, 50, 55, Yes, Yes], [[a0, a2, a3, s], 50, 0, 50, 55, Yes, Yes], [[a1, a2, a3, s], 50, 0, 50, 55, Yes, Yes], [[a0, a1, a2, a3, s], 50, 0, 50, 55, Yes, Yes], [[a4, s], 50, 0, 50, 55, Yes, Yes], [[a0, a4, s], 50, 0, 50, 55, Yes, Yes], [[a1, a4, s], 50, 0, 50, 55, Yes, Yes], [[a0, a1, a4, s], 50, 0, 50, 55, Yes, Yes], [[a2, a4, s], 50, 0, 50, 55, Yes, Yes], [[a0, a2, a4, s], 50, 0, 50, 55, Yes, Yes], [[a1, a2, a4, s], 50, 0, 50, 55, Yes, Yes], [[a0, a1, a2, a4, s], 50, 0, 50, 55, Yes, Yes], [[a3, a4, s], 50, 0, 50, 55, Yes, Yes], [[a0, a3, a4, s], 50, 0, 50, 55, Yes, Yes], [[a1, a3, a4, s], 50, 0, 50, 55, Yes, Yes], [[a0, a1, a3, a4, s], 50, 0, 50, 55, Yes, Yes], [[a2, a3, a4, s], 50, 0, 50, 55, Yes, Yes], [[a0, a2, a3, a4, s], 50, 0, 50, 55, Yes, Yes], [[a1, a2, a3, a4, s], 50, 0, 50, 55, Yes, Yes], [[a0, a1, a2, a3, a4, s], 50, 0, 50, 55, Yes, Yes]], Yes, No, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum).
Reviewed step-by-step derivation
Max-flow min-cut theorem · qf-mod46-mfmc-flow-cut-upper-bound · candidate
34. A Feasible Flow Is Bounded above by Every Cut, Not Automatically Maximum
Use complete directed vertices=[s, a0, a1, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t]], source=s, sink=t, aligned nonnegative exact capacities=[10, 10, 11, 11], candidate flows=[9, 9, 10, 10]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (source_value, sink_value, flow_cut_rows, all_cut_bounds_hold, automatic_maximum_claim, certificate).
Cut capacity counts outgoing edges only; net flow subtracts incoming flow. No automatic_maximum_claim does not classify this particular flow as necessarily nonmaximum.
View answer and explanation
(19, 19, [[[s], 19, 0, 19, 21, Yes, Yes], [[a0, s], 19, 0, 19, 21, Yes, Yes], [[a1, s], 19, 0, 19, 21, Yes, Yes], [[a0, a1, s], 19, 0, 19, 21, Yes, Yes]], Yes, No, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum)
Require nonnegative capacity-bounded flows and conservation at every internal vertex. For every cut compute outgoing flow, incoming flow, their difference, capacity, net/source-value equality and the upper bound. The fixed No withholds automatic maximality; it does not assert that this particular feasible flow is nonmaximum. The exact result, retaining the requested order, is (19, 19, [[[s], 19, 0, 19, 21, Yes, Yes], [[a0, s], 19, 0, 19, 21, Yes, Yes], [[a1, s], 19, 0, 19, 21, Yes, Yes], [[a0, a1, s], 19, 0, 19, 21, Yes, Yes]], Yes, No, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum).
Reviewed step-by-step derivation
Max-flow min-cut theorem · qf-mod46-mfmc-flow-cut-upper-bound · candidate
35. A Feasible Flow Is Bounded above by Every Cut, Not Automatically Maximum
Use complete directed vertices=[s, a0, a1, a2, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t], [s, a2], [a2, t], [a0, s]], source=s, sink=t, aligned nonnegative exact capacities=[11, 11, 12, 12, 13, 13, 16], candidate flows=[10, 10, 11, 11, 12, 12, 0]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (source_value, sink_value, flow_cut_rows, all_cut_bounds_hold, automatic_maximum_claim, certificate).
Cut capacity counts outgoing edges only; net flow subtracts incoming flow. No automatic_maximum_claim does not classify this particular flow as necessarily nonmaximum.
View answer and explanation
(33, 33, [[[s], 33, 0, 33, 36, Yes, Yes], [[a0, s], 33, 0, 33, 36, Yes, Yes], [[a1, s], 33, 0, 33, 36, Yes, Yes], [[a0, a1, s], 33, 0, 33, 36, Yes, Yes], [[a2, s], 33, 0, 33, 36, Yes, Yes], [[a0, a2, s], 33, 0, 33, 36, Yes, Yes], [[a1, a2, s], 33, 0, 33, 36, Yes, Yes], [[a0, a1, a2, s], 33, 0, 33, 36, Yes, Yes]], Yes, No, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum)
Require nonnegative capacity-bounded flows and conservation at every internal vertex. For every cut compute outgoing flow, incoming flow, their difference, capacity, net/source-value equality and the upper bound. The fixed No withholds automatic maximality; it does not assert that this particular feasible flow is nonmaximum. The exact result, retaining the requested order, is (33, 33, [[[s], 33, 0, 33, 36, Yes, Yes], [[a0, s], 33, 0, 33, 36, Yes, Yes], [[a1, s], 33, 0, 33, 36, Yes, Yes], [[a0, a1, s], 33, 0, 33, 36, Yes, Yes], [[a2, s], 33, 0, 33, 36, Yes, Yes], [[a0, a2, s], 33, 0, 33, 36, Yes, Yes], [[a1, a2, s], 33, 0, 33, 36, Yes, Yes], [[a0, a1, a2, s], 33, 0, 33, 36, Yes, Yes]], Yes, No, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum).
Reviewed step-by-step derivation
Max-flow min-cut theorem · qf-mod46-mfmc-flow-cut-upper-bound · candidate
36. A Feasible Flow Is Bounded above by Every Cut, Not Automatically Maximum
Use complete directed vertices=[s, a0, a1, a2, a3, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t], [s, a2], [a2, t], [s, a3], [a3, t]], source=s, sink=t, aligned nonnegative exact capacities=[12, 12, 13, 13, 14, 14, 15, 15], candidate flows=[11, 11, 12, 12, 13, 13, 14, 14]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (source_value, sink_value, flow_cut_rows, all_cut_bounds_hold, automatic_maximum_claim, certificate).
Cut capacity counts outgoing edges only; net flow subtracts incoming flow. No automatic_maximum_claim does not classify this particular flow as necessarily nonmaximum.
View answer and explanation
(50, 50, [[[s], 50, 0, 50, 54, Yes, Yes], [[a0, s], 50, 0, 50, 54, Yes, Yes], [[a1, s], 50, 0, 50, 54, Yes, Yes], [[a0, a1, s], 50, 0, 50, 54, Yes, Yes], [[a2, s], 50, 0, 50, 54, Yes, Yes], [[a0, a2, s], 50, 0, 50, 54, Yes, Yes], [[a1, a2, s], 50, 0, 50, 54, Yes, Yes], [[a0, a1, a2, s], 50, 0, 50, 54, Yes, Yes], [[a3, s], 50, 0, 50, 54, Yes, Yes], [[a0, a3, s], 50, 0, 50, 54, Yes, Yes], [[a1, a3, s], 50, 0, 50, 54, Yes, Yes], [[a0, a1, a3, s], 50, 0, 50, 54, Yes, Yes], [[a2, a3, s], 50, 0, 50, 54, Yes, Yes], [[a0, a2, a3, s], 50, 0, 50, 54, Yes, Yes], [[a1, a2, a3, s], 50, 0, 50, 54, Yes, Yes], [[a0, a1, a2, a3, s], 50, 0, 50, 54, Yes, Yes]], Yes, No, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum)
Require nonnegative capacity-bounded flows and conservation at every internal vertex. For every cut compute outgoing flow, incoming flow, their difference, capacity, net/source-value equality and the upper bound. The fixed No withholds automatic maximality; it does not assert that this particular feasible flow is nonmaximum. The exact result, retaining the requested order, is (50, 50, [[[s], 50, 0, 50, 54, Yes, Yes], [[a0, s], 50, 0, 50, 54, Yes, Yes], [[a1, s], 50, 0, 50, 54, Yes, Yes], [[a0, a1, s], 50, 0, 50, 54, Yes, Yes], [[a2, s], 50, 0, 50, 54, Yes, Yes], [[a0, a2, s], 50, 0, 50, 54, Yes, Yes], [[a1, a2, s], 50, 0, 50, 54, Yes, Yes], [[a0, a1, a2, s], 50, 0, 50, 54, Yes, Yes], [[a3, s], 50, 0, 50, 54, Yes, Yes], [[a0, a3, s], 50, 0, 50, 54, Yes, Yes], [[a1, a3, s], 50, 0, 50, 54, Yes, Yes], [[a0, a1, a3, s], 50, 0, 50, 54, Yes, Yes], [[a2, a3, s], 50, 0, 50, 54, Yes, Yes], [[a0, a2, a3, s], 50, 0, 50, 54, Yes, Yes], [[a1, a2, a3, s], 50, 0, 50, 54, Yes, Yes], [[a0, a1, a2, a3, s], 50, 0, 50, 54, Yes, Yes]], Yes, No, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum).
Reviewed step-by-step derivation
Max-flow min-cut theorem · qf-mod46-mfmc-flow-cut-upper-bound · candidate
37. A Feasible Flow Is Bounded above by Every Cut, Not Automatically Maximum
Use complete directed vertices=[s, a0, a1, a2, a3, a4, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t], [s, a2], [a2, t], [s, a3], [a3, t], [s, a4], [a4, t]], source=s, sink=t, aligned nonnegative exact capacities=[13, 13, 14, 14, 15, 15, 16, 16, 17, 17], candidate flows=[12, 12, 13, 13, 14, 14, 15, 15, 16, 16]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (source_value, sink_value, flow_cut_rows, all_cut_bounds_hold, automatic_maximum_claim, certificate).
Cut capacity counts outgoing edges only; net flow subtracts incoming flow. No automatic_maximum_claim does not classify this particular flow as necessarily nonmaximum.
View answer and explanation
(70, 70, [[[s], 70, 0, 70, 75, Yes, Yes], [[a0, s], 70, 0, 70, 75, Yes, Yes], [[a1, s], 70, 0, 70, 75, Yes, Yes], [[a0, a1, s], 70, 0, 70, 75, Yes, Yes], [[a2, s], 70, 0, 70, 75, Yes, Yes], [[a0, a2, s], 70, 0, 70, 75, Yes, Yes], [[a1, a2, s], 70, 0, 70, 75, Yes, Yes], [[a0, a1, a2, s], 70, 0, 70, 75, Yes, Yes], [[a3, s], 70, 0, 70, 75, Yes, Yes], [[a0, a3, s], 70, 0, 70, 75, Yes, Yes], [[a1, a3, s], 70, 0, 70, 75, Yes, Yes], [[a0, a1, a3, s], 70, 0, 70, 75, Yes, Yes], [[a2, a3, s], 70, 0, 70, 75, Yes, Yes], [[a0, a2, a3, s], 70, 0, 70, 75, Yes, Yes], [[a1, a2, a3, s], 70, 0, 70, 75, Yes, Yes], [[a0, a1, a2, a3, s], 70, 0, 70, 75, Yes, Yes], [[a4, s], 70, 0, 70, 75, Yes, Yes], [[a0, a4, s], 70, 0, 70, 75, Yes, Yes], [[a1, a4, s], 70, 0, 70, 75, Yes, Yes], [[a0, a1, a4, s], 70, 0, 70, 75, Yes, Yes], [[a2, a4, s], 70, 0, 70, 75, Yes, Yes], [[a0, a2, a4, s], 70, 0, 70, 75, Yes, Yes], [[a1, a2, a4, s], 70, 0, 70, 75, Yes, Yes], [[a0, a1, a2, a4, s], 70, 0, 70, 75, Yes, Yes], [[a3, a4, s], 70, 0, 70, 75, Yes, Yes], [[a0, a3, a4, s], 70, 0, 70, 75, Yes, Yes], [[a1, a3, a4, s], 70, 0, 70, 75, Yes, Yes], [[a0, a1, a3, a4, s], 70, 0, 70, 75, Yes, Yes], [[a2, a3, a4, s], 70, 0, 70, 75, Yes, Yes], [[a0, a2, a3, a4, s], 70, 0, 70, 75, Yes, Yes], [[a1, a2, a3, a4, s], 70, 0, 70, 75, Yes, Yes], [[a0, a1, a2, a3, a4, s], 70, 0, 70, 75, Yes, Yes]], Yes, No, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum)
Require nonnegative capacity-bounded flows and conservation at every internal vertex. For every cut compute outgoing flow, incoming flow, their difference, capacity, net/source-value equality and the upper bound. The fixed No withholds automatic maximality; it does not assert that this particular feasible flow is nonmaximum. The exact result, retaining the requested order, is (70, 70, [[[s], 70, 0, 70, 75, Yes, Yes], [[a0, s], 70, 0, 70, 75, Yes, Yes], [[a1, s], 70, 0, 70, 75, Yes, Yes], [[a0, a1, s], 70, 0, 70, 75, Yes, Yes], [[a2, s], 70, 0, 70, 75, Yes, Yes], [[a0, a2, s], 70, 0, 70, 75, Yes, Yes], [[a1, a2, s], 70, 0, 70, 75, Yes, Yes], [[a0, a1, a2, s], 70, 0, 70, 75, Yes, Yes], [[a3, s], 70, 0, 70, 75, Yes, Yes], [[a0, a3, s], 70, 0, 70, 75, Yes, Yes], [[a1, a3, s], 70, 0, 70, 75, Yes, Yes], [[a0, a1, a3, s], 70, 0, 70, 75, Yes, Yes], [[a2, a3, s], 70, 0, 70, 75, Yes, Yes], [[a0, a2, a3, s], 70, 0, 70, 75, Yes, Yes], [[a1, a2, a3, s], 70, 0, 70, 75, Yes, Yes], [[a0, a1, a2, a3, s], 70, 0, 70, 75, Yes, Yes], [[a4, s], 70, 0, 70, 75, Yes, Yes], [[a0, a4, s], 70, 0, 70, 75, Yes, Yes], [[a1, a4, s], 70, 0, 70, 75, Yes, Yes], [[a0, a1, a4, s], 70, 0, 70, 75, Yes, Yes], [[a2, a4, s], 70, 0, 70, 75, Yes, Yes], [[a0, a2, a4, s], 70, 0, 70, 75, Yes, Yes], [[a1, a2, a4, s], 70, 0, 70, 75, Yes, Yes], [[a0, a1, a2, a4, s], 70, 0, 70, 75, Yes, Yes], [[a3, a4, s], 70, 0, 70, 75, Yes, Yes], [[a0, a3, a4, s], 70, 0, 70, 75, Yes, Yes], [[a1, a3, a4, s], 70, 0, 70, 75, Yes, Yes], [[a0, a1, a3, a4, s], 70, 0, 70, 75, Yes, Yes], [[a2, a3, a4, s], 70, 0, 70, 75, Yes, Yes], [[a0, a2, a3, a4, s], 70, 0, 70, 75, Yes, Yes], [[a1, a2, a3, a4, s], 70, 0, 70, 75, Yes, Yes], [[a0, a1, a2, a3, a4, s], 70, 0, 70, 75, Yes, Yes]], Yes, No, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum).
Reviewed step-by-step derivation
Max-flow min-cut theorem · qf-mod46-mfmc-flow-cut-upper-bound · candidate
38. A Feasible Flow Is Bounded above by Every Cut, Not Automatically Maximum
Use complete directed vertices=[s, a0, a1, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t], [a0, s]], source=s, sink=t, aligned nonnegative exact capacities=[14, 14, 15, 15, 19], candidate flows=[13, 13, 14, 14, 0]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (source_value, sink_value, flow_cut_rows, all_cut_bounds_hold, automatic_maximum_claim, certificate).
Cut capacity counts outgoing edges only; net flow subtracts incoming flow. No automatic_maximum_claim does not classify this particular flow as necessarily nonmaximum.
View answer and explanation
(27, 27, [[[s], 27, 0, 27, 29, Yes, Yes], [[a0, s], 27, 0, 27, 29, Yes, Yes], [[a1, s], 27, 0, 27, 29, Yes, Yes], [[a0, a1, s], 27, 0, 27, 29, Yes, Yes]], Yes, No, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum)
Require nonnegative capacity-bounded flows and conservation at every internal vertex. For every cut compute outgoing flow, incoming flow, their difference, capacity, net/source-value equality and the upper bound. The fixed No withholds automatic maximality; it does not assert that this particular feasible flow is nonmaximum. The exact result, retaining the requested order, is (27, 27, [[[s], 27, 0, 27, 29, Yes, Yes], [[a0, s], 27, 0, 27, 29, Yes, Yes], [[a1, s], 27, 0, 27, 29, Yes, Yes], [[a0, a1, s], 27, 0, 27, 29, Yes, Yes]], Yes, No, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum).
Reviewed step-by-step derivation
Max-flow min-cut theorem · qf-mod46-mfmc-flow-cut-upper-bound · candidate
39. A Feasible Flow Is Bounded above by Every Cut, Not Automatically Maximum
Use complete directed vertices=[s, a0, a1, a2, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t], [s, a2], [a2, t]], source=s, sink=t, aligned nonnegative exact capacities=[15, 15, 16, 16, 17, 17], candidate flows=[14, 14, 15, 15, 16, 16]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (source_value, sink_value, flow_cut_rows, all_cut_bounds_hold, automatic_maximum_claim, certificate).
Cut capacity counts outgoing edges only; net flow subtracts incoming flow. No automatic_maximum_claim does not classify this particular flow as necessarily nonmaximum.
View answer and explanation
(45, 45, [[[s], 45, 0, 45, 48, Yes, Yes], [[a0, s], 45, 0, 45, 48, Yes, Yes], [[a1, s], 45, 0, 45, 48, Yes, Yes], [[a0, a1, s], 45, 0, 45, 48, Yes, Yes], [[a2, s], 45, 0, 45, 48, Yes, Yes], [[a0, a2, s], 45, 0, 45, 48, Yes, Yes], [[a1, a2, s], 45, 0, 45, 48, Yes, Yes], [[a0, a1, a2, s], 45, 0, 45, 48, Yes, Yes]], Yes, No, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum)
Require nonnegative capacity-bounded flows and conservation at every internal vertex. For every cut compute outgoing flow, incoming flow, their difference, capacity, net/source-value equality and the upper bound. The fixed No withholds automatic maximality; it does not assert that this particular feasible flow is nonmaximum. The exact result, retaining the requested order, is (45, 45, [[[s], 45, 0, 45, 48, Yes, Yes], [[a0, s], 45, 0, 45, 48, Yes, Yes], [[a1, s], 45, 0, 45, 48, Yes, Yes], [[a0, a1, s], 45, 0, 45, 48, Yes, Yes], [[a2, s], 45, 0, 45, 48, Yes, Yes], [[a0, a2, s], 45, 0, 45, 48, Yes, Yes], [[a1, a2, s], 45, 0, 45, 48, Yes, Yes], [[a0, a1, a2, s], 45, 0, 45, 48, Yes, Yes]], Yes, No, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum).
Reviewed step-by-step derivation
Max-flow min-cut theorem · qf-mod46-mfmc-flow-cut-upper-bound · candidate
40. A Feasible Flow Is Bounded above by Every Cut, Not Automatically Maximum
Use complete directed vertices=[s, a0, a1, a2, a3, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t], [s, a2], [a2, t], [s, a3], [a3, t]], source=s, sink=t, aligned nonnegative exact capacities=[16, 16, 17, 17, 18, 18, 19, 19], candidate flows=[15, 15, 16, 16, 17, 17, 18, 18]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (source_value, sink_value, flow_cut_rows, all_cut_bounds_hold, automatic_maximum_claim, certificate).
Cut capacity counts outgoing edges only; net flow subtracts incoming flow. No automatic_maximum_claim does not classify this particular flow as necessarily nonmaximum.
View answer and explanation
(66, 66, [[[s], 66, 0, 66, 70, Yes, Yes], [[a0, s], 66, 0, 66, 70, Yes, Yes], [[a1, s], 66, 0, 66, 70, Yes, Yes], [[a0, a1, s], 66, 0, 66, 70, Yes, Yes], [[a2, s], 66, 0, 66, 70, Yes, Yes], [[a0, a2, s], 66, 0, 66, 70, Yes, Yes], [[a1, a2, s], 66, 0, 66, 70, Yes, Yes], [[a0, a1, a2, s], 66, 0, 66, 70, Yes, Yes], [[a3, s], 66, 0, 66, 70, Yes, Yes], [[a0, a3, s], 66, 0, 66, 70, Yes, Yes], [[a1, a3, s], 66, 0, 66, 70, Yes, Yes], [[a0, a1, a3, s], 66, 0, 66, 70, Yes, Yes], [[a2, a3, s], 66, 0, 66, 70, Yes, Yes], [[a0, a2, a3, s], 66, 0, 66, 70, Yes, Yes], [[a1, a2, a3, s], 66, 0, 66, 70, Yes, Yes], [[a0, a1, a2, a3, s], 66, 0, 66, 70, Yes, Yes]], Yes, No, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum)
Require nonnegative capacity-bounded flows and conservation at every internal vertex. For every cut compute outgoing flow, incoming flow, their difference, capacity, net/source-value equality and the upper bound. The fixed No withholds automatic maximality; it does not assert that this particular feasible flow is nonmaximum. The exact result, retaining the requested order, is (66, 66, [[[s], 66, 0, 66, 70, Yes, Yes], [[a0, s], 66, 0, 66, 70, Yes, Yes], [[a1, s], 66, 0, 66, 70, Yes, Yes], [[a0, a1, s], 66, 0, 66, 70, Yes, Yes], [[a2, s], 66, 0, 66, 70, Yes, Yes], [[a0, a2, s], 66, 0, 66, 70, Yes, Yes], [[a1, a2, s], 66, 0, 66, 70, Yes, Yes], [[a0, a1, a2, s], 66, 0, 66, 70, Yes, Yes], [[a3, s], 66, 0, 66, 70, Yes, Yes], [[a0, a3, s], 66, 0, 66, 70, Yes, Yes], [[a1, a3, s], 66, 0, 66, 70, Yes, Yes], [[a0, a1, a3, s], 66, 0, 66, 70, Yes, Yes], [[a2, a3, s], 66, 0, 66, 70, Yes, Yes], [[a0, a2, a3, s], 66, 0, 66, 70, Yes, Yes], [[a1, a2, a3, s], 66, 0, 66, 70, Yes, Yes], [[a0, a1, a2, a3, s], 66, 0, 66, 70, Yes, Yes]], Yes, No, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum).
Reviewed step-by-step derivation
Max-flow min-cut theorem · qf-mod46-mfmc-flow-cut-upper-bound · candidate
41. A Feasible Flow Is Bounded above by Every Cut, Not Automatically Maximum
Use complete directed vertices=[s, a0, a1, a2, a3, a4, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t], [s, a2], [a2, t], [s, a3], [a3, t], [s, a4], [a4, t], [a0, s]], source=s, sink=t, aligned nonnegative exact capacities=[17, 17, 18, 18, 19, 19, 20, 20, 21, 21, 22], candidate flows=[16, 16, 17, 17, 18, 18, 19, 19, 20, 20, 0]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (source_value, sink_value, flow_cut_rows, all_cut_bounds_hold, automatic_maximum_claim, certificate).
Cut capacity counts outgoing edges only; net flow subtracts incoming flow. No automatic_maximum_claim does not classify this particular flow as necessarily nonmaximum.
View answer and explanation
(90, 90, [[[s], 90, 0, 90, 95, Yes, Yes], [[a0, s], 90, 0, 90, 95, Yes, Yes], [[a1, s], 90, 0, 90, 95, Yes, Yes], [[a0, a1, s], 90, 0, 90, 95, Yes, Yes], [[a2, s], 90, 0, 90, 95, Yes, Yes], [[a0, a2, s], 90, 0, 90, 95, Yes, Yes], [[a1, a2, s], 90, 0, 90, 95, Yes, Yes], [[a0, a1, a2, s], 90, 0, 90, 95, Yes, Yes], [[a3, s], 90, 0, 90, 95, Yes, Yes], [[a0, a3, s], 90, 0, 90, 95, Yes, Yes], [[a1, a3, s], 90, 0, 90, 95, Yes, Yes], [[a0, a1, a3, s], 90, 0, 90, 95, Yes, Yes], [[a2, a3, s], 90, 0, 90, 95, Yes, Yes], [[a0, a2, a3, s], 90, 0, 90, 95, Yes, Yes], [[a1, a2, a3, s], 90, 0, 90, 95, Yes, Yes], [[a0, a1, a2, a3, s], 90, 0, 90, 95, Yes, Yes], [[a4, s], 90, 0, 90, 95, Yes, Yes], [[a0, a4, s], 90, 0, 90, 95, Yes, Yes], [[a1, a4, s], 90, 0, 90, 95, Yes, Yes], [[a0, a1, a4, s], 90, 0, 90, 95, Yes, Yes], [[a2, a4, s], 90, 0, 90, 95, Yes, Yes], [[a0, a2, a4, s], 90, 0, 90, 95, Yes, Yes], [[a1, a2, a4, s], 90, 0, 90, 95, Yes, Yes], [[a0, a1, a2, a4, s], 90, 0, 90, 95, Yes, Yes], [[a3, a4, s], 90, 0, 90, 95, Yes, Yes], [[a0, a3, a4, s], 90, 0, 90, 95, Yes, Yes], [[a1, a3, a4, s], 90, 0, 90, 95, Yes, Yes], [[a0, a1, a3, a4, s], 90, 0, 90, 95, Yes, Yes], [[a2, a3, a4, s], 90, 0, 90, 95, Yes, Yes], [[a0, a2, a3, a4, s], 90, 0, 90, 95, Yes, Yes], [[a1, a2, a3, a4, s], 90, 0, 90, 95, Yes, Yes], [[a0, a1, a2, a3, a4, s], 90, 0, 90, 95, Yes, Yes]], Yes, No, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum)
Require nonnegative capacity-bounded flows and conservation at every internal vertex. For every cut compute outgoing flow, incoming flow, their difference, capacity, net/source-value equality and the upper bound. The fixed No withholds automatic maximality; it does not assert that this particular feasible flow is nonmaximum. The exact result, retaining the requested order, is (90, 90, [[[s], 90, 0, 90, 95, Yes, Yes], [[a0, s], 90, 0, 90, 95, Yes, Yes], [[a1, s], 90, 0, 90, 95, Yes, Yes], [[a0, a1, s], 90, 0, 90, 95, Yes, Yes], [[a2, s], 90, 0, 90, 95, Yes, Yes], [[a0, a2, s], 90, 0, 90, 95, Yes, Yes], [[a1, a2, s], 90, 0, 90, 95, Yes, Yes], [[a0, a1, a2, s], 90, 0, 90, 95, Yes, Yes], [[a3, s], 90, 0, 90, 95, Yes, Yes], [[a0, a3, s], 90, 0, 90, 95, Yes, Yes], [[a1, a3, s], 90, 0, 90, 95, Yes, Yes], [[a0, a1, a3, s], 90, 0, 90, 95, Yes, Yes], [[a2, a3, s], 90, 0, 90, 95, Yes, Yes], [[a0, a2, a3, s], 90, 0, 90, 95, Yes, Yes], [[a1, a2, a3, s], 90, 0, 90, 95, Yes, Yes], [[a0, a1, a2, a3, s], 90, 0, 90, 95, Yes, Yes], [[a4, s], 90, 0, 90, 95, Yes, Yes], [[a0, a4, s], 90, 0, 90, 95, Yes, Yes], [[a1, a4, s], 90, 0, 90, 95, Yes, Yes], [[a0, a1, a4, s], 90, 0, 90, 95, Yes, Yes], [[a2, a4, s], 90, 0, 90, 95, Yes, Yes], [[a0, a2, a4, s], 90, 0, 90, 95, Yes, Yes], [[a1, a2, a4, s], 90, 0, 90, 95, Yes, Yes], [[a0, a1, a2, a4, s], 90, 0, 90, 95, Yes, Yes], [[a3, a4, s], 90, 0, 90, 95, Yes, Yes], [[a0, a3, a4, s], 90, 0, 90, 95, Yes, Yes], [[a1, a3, a4, s], 90, 0, 90, 95, Yes, Yes], [[a0, a1, a3, a4, s], 90, 0, 90, 95, Yes, Yes], [[a2, a3, a4, s], 90, 0, 90, 95, Yes, Yes], [[a0, a2, a3, a4, s], 90, 0, 90, 95, Yes, Yes], [[a1, a2, a3, a4, s], 90, 0, 90, 95, Yes, Yes], [[a0, a1, a2, a3, a4, s], 90, 0, 90, 95, Yes, Yes]], Yes, No, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum).
Reviewed step-by-step derivation
Max-flow min-cut theorem · qf-mod46-mfmc-flow-cut-upper-bound · candidate
42. A Feasible Flow Is Bounded above by Every Cut, Not Automatically Maximum
Use complete directed vertices=[s, a0, a1, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t]], source=s, sink=t, aligned nonnegative exact capacities=[18, 18, 19, 19], candidate flows=[17, 17, 18, 18]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (source_value, sink_value, flow_cut_rows, all_cut_bounds_hold, automatic_maximum_claim, certificate).
Cut capacity counts outgoing edges only; net flow subtracts incoming flow. No automatic_maximum_claim does not classify this particular flow as necessarily nonmaximum.
View answer and explanation
(35, 35, [[[s], 35, 0, 35, 37, Yes, Yes], [[a0, s], 35, 0, 35, 37, Yes, Yes], [[a1, s], 35, 0, 35, 37, Yes, Yes], [[a0, a1, s], 35, 0, 35, 37, Yes, Yes]], Yes, No, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum)
Require nonnegative capacity-bounded flows and conservation at every internal vertex. For every cut compute outgoing flow, incoming flow, their difference, capacity, net/source-value equality and the upper bound. The fixed No withholds automatic maximality; it does not assert that this particular feasible flow is nonmaximum. The exact result, retaining the requested order, is (35, 35, [[[s], 35, 0, 35, 37, Yes, Yes], [[a0, s], 35, 0, 35, 37, Yes, Yes], [[a1, s], 35, 0, 35, 37, Yes, Yes], [[a0, a1, s], 35, 0, 35, 37, Yes, Yes]], Yes, No, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum).
Reviewed step-by-step derivation
Max-flow min-cut theorem · qf-mod46-mfmc-flow-cut-upper-bound · candidate
43. A Feasible Flow Is Bounded above by Every Cut, Not Automatically Maximum
Use complete directed vertices=[s, a0, a1, a2, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t], [s, a2], [a2, t]], source=s, sink=t, aligned nonnegative exact capacities=[19, 19, 20, 20, 21, 21], candidate flows=[18, 18, 19, 19, 20, 20]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (source_value, sink_value, flow_cut_rows, all_cut_bounds_hold, automatic_maximum_claim, certificate).
Cut capacity counts outgoing edges only; net flow subtracts incoming flow. No automatic_maximum_claim does not classify this particular flow as necessarily nonmaximum.
View answer and explanation
(57, 57, [[[s], 57, 0, 57, 60, Yes, Yes], [[a0, s], 57, 0, 57, 60, Yes, Yes], [[a1, s], 57, 0, 57, 60, Yes, Yes], [[a0, a1, s], 57, 0, 57, 60, Yes, Yes], [[a2, s], 57, 0, 57, 60, Yes, Yes], [[a0, a2, s], 57, 0, 57, 60, Yes, Yes], [[a1, a2, s], 57, 0, 57, 60, Yes, Yes], [[a0, a1, a2, s], 57, 0, 57, 60, Yes, Yes]], Yes, No, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum)
Require nonnegative capacity-bounded flows and conservation at every internal vertex. For every cut compute outgoing flow, incoming flow, their difference, capacity, net/source-value equality and the upper bound. The fixed No withholds automatic maximality; it does not assert that this particular feasible flow is nonmaximum. The exact result, retaining the requested order, is (57, 57, [[[s], 57, 0, 57, 60, Yes, Yes], [[a0, s], 57, 0, 57, 60, Yes, Yes], [[a1, s], 57, 0, 57, 60, Yes, Yes], [[a0, a1, s], 57, 0, 57, 60, Yes, Yes], [[a2, s], 57, 0, 57, 60, Yes, Yes], [[a0, a2, s], 57, 0, 57, 60, Yes, Yes], [[a1, a2, s], 57, 0, 57, 60, Yes, Yes], [[a0, a1, a2, s], 57, 0, 57, 60, Yes, Yes]], Yes, No, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum).
Reviewed step-by-step derivation
Max-flow min-cut theorem · qf-mod46-mfmc-flow-cut-upper-bound · candidate
44. A Feasible Flow Is Bounded above by Every Cut, Not Automatically Maximum
Use complete directed vertices=[s, a0, a1, a2, a3, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t], [s, a2], [a2, t], [s, a3], [a3, t], [a0, s]], source=s, sink=t, aligned nonnegative exact capacities=[20, 20, 21, 21, 22, 22, 23, 23, 25], candidate flows=[19, 19, 20, 20, 21, 21, 22, 22, 0]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (source_value, sink_value, flow_cut_rows, all_cut_bounds_hold, automatic_maximum_claim, certificate).
Cut capacity counts outgoing edges only; net flow subtracts incoming flow. No automatic_maximum_claim does not classify this particular flow as necessarily nonmaximum.
View answer and explanation
(82, 82, [[[s], 82, 0, 82, 86, Yes, Yes], [[a0, s], 82, 0, 82, 86, Yes, Yes], [[a1, s], 82, 0, 82, 86, Yes, Yes], [[a0, a1, s], 82, 0, 82, 86, Yes, Yes], [[a2, s], 82, 0, 82, 86, Yes, Yes], [[a0, a2, s], 82, 0, 82, 86, Yes, Yes], [[a1, a2, s], 82, 0, 82, 86, Yes, Yes], [[a0, a1, a2, s], 82, 0, 82, 86, Yes, Yes], [[a3, s], 82, 0, 82, 86, Yes, Yes], [[a0, a3, s], 82, 0, 82, 86, Yes, Yes], [[a1, a3, s], 82, 0, 82, 86, Yes, Yes], [[a0, a1, a3, s], 82, 0, 82, 86, Yes, Yes], [[a2, a3, s], 82, 0, 82, 86, Yes, Yes], [[a0, a2, a3, s], 82, 0, 82, 86, Yes, Yes], [[a1, a2, a3, s], 82, 0, 82, 86, Yes, Yes], [[a0, a1, a2, a3, s], 82, 0, 82, 86, Yes, Yes]], Yes, No, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum)
Require nonnegative capacity-bounded flows and conservation at every internal vertex. For every cut compute outgoing flow, incoming flow, their difference, capacity, net/source-value equality and the upper bound. The fixed No withholds automatic maximality; it does not assert that this particular feasible flow is nonmaximum. The exact result, retaining the requested order, is (82, 82, [[[s], 82, 0, 82, 86, Yes, Yes], [[a0, s], 82, 0, 82, 86, Yes, Yes], [[a1, s], 82, 0, 82, 86, Yes, Yes], [[a0, a1, s], 82, 0, 82, 86, Yes, Yes], [[a2, s], 82, 0, 82, 86, Yes, Yes], [[a0, a2, s], 82, 0, 82, 86, Yes, Yes], [[a1, a2, s], 82, 0, 82, 86, Yes, Yes], [[a0, a1, a2, s], 82, 0, 82, 86, Yes, Yes], [[a3, s], 82, 0, 82, 86, Yes, Yes], [[a0, a3, s], 82, 0, 82, 86, Yes, Yes], [[a1, a3, s], 82, 0, 82, 86, Yes, Yes], [[a0, a1, a3, s], 82, 0, 82, 86, Yes, Yes], [[a2, a3, s], 82, 0, 82, 86, Yes, Yes], [[a0, a2, a3, s], 82, 0, 82, 86, Yes, Yes], [[a1, a2, a3, s], 82, 0, 82, 86, Yes, Yes], [[a0, a1, a2, a3, s], 82, 0, 82, 86, Yes, Yes]], Yes, No, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum).
Reviewed step-by-step derivation
Max-flow min-cut theorem · qf-mod46-mfmc-flow-cut-upper-bound · candidate
45. A Feasible Flow Is Bounded above by Every Cut, Not Automatically Maximum
Use complete directed vertices=[s, a0, a1, a2, a3, a4, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t], [s, a2], [a2, t], [s, a3], [a3, t], [s, a4], [a4, t]], source=s, sink=t, aligned nonnegative exact capacities=[21, 21, 22, 22, 23, 23, 24, 24, 25, 25], candidate flows=[20, 20, 21, 21, 22, 22, 23, 23, 24, 24]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (source_value, sink_value, flow_cut_rows, all_cut_bounds_hold, automatic_maximum_claim, certificate).
Cut capacity counts outgoing edges only; net flow subtracts incoming flow. No automatic_maximum_claim does not classify this particular flow as necessarily nonmaximum.
View answer and explanation
(110, 110, [[[s], 110, 0, 110, 115, Yes, Yes], [[a0, s], 110, 0, 110, 115, Yes, Yes], [[a1, s], 110, 0, 110, 115, Yes, Yes], [[a0, a1, s], 110, 0, 110, 115, Yes, Yes], [[a2, s], 110, 0, 110, 115, Yes, Yes], [[a0, a2, s], 110, 0, 110, 115, Yes, Yes], [[a1, a2, s], 110, 0, 110, 115, Yes, Yes], [[a0, a1, a2, s], 110, 0, 110, 115, Yes, Yes], [[a3, s], 110, 0, 110, 115, Yes, Yes], [[a0, a3, s], 110, 0, 110, 115, Yes, Yes], [[a1, a3, s], 110, 0, 110, 115, Yes, Yes], [[a0, a1, a3, s], 110, 0, 110, 115, Yes, Yes], [[a2, a3, s], 110, 0, 110, 115, Yes, Yes], [[a0, a2, a3, s], 110, 0, 110, 115, Yes, Yes], [[a1, a2, a3, s], 110, 0, 110, 115, Yes, Yes], [[a0, a1, a2, a3, s], 110, 0, 110, 115, Yes, Yes], [[a4, s], 110, 0, 110, 115, Yes, Yes], [[a0, a4, s], 110, 0, 110, 115, Yes, Yes], [[a1, a4, s], 110, 0, 110, 115, Yes, Yes], [[a0, a1, a4, s], 110, 0, 110, 115, Yes, Yes], [[a2, a4, s], 110, 0, 110, 115, Yes, Yes], [[a0, a2, a4, s], 110, 0, 110, 115, Yes, Yes], [[a1, a2, a4, s], 110, 0, 110, 115, Yes, Yes], [[a0, a1, a2, a4, s], 110, 0, 110, 115, Yes, Yes], [[a3, a4, s], 110, 0, 110, 115, Yes, Yes], [[a0, a3, a4, s], 110, 0, 110, 115, Yes, Yes], [[a1, a3, a4, s], 110, 0, 110, 115, Yes, Yes], [[a0, a1, a3, a4, s], 110, 0, 110, 115, Yes, Yes], [[a2, a3, a4, s], 110, 0, 110, 115, Yes, Yes], [[a0, a2, a3, a4, s], 110, 0, 110, 115, Yes, Yes], [[a1, a2, a3, a4, s], 110, 0, 110, 115, Yes, Yes], [[a0, a1, a2, a3, a4, s], 110, 0, 110, 115, Yes, Yes]], Yes, No, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum)
Require nonnegative capacity-bounded flows and conservation at every internal vertex. For every cut compute outgoing flow, incoming flow, their difference, capacity, net/source-value equality and the upper bound. The fixed No withholds automatic maximality; it does not assert that this particular feasible flow is nonmaximum. The exact result, retaining the requested order, is (110, 110, [[[s], 110, 0, 110, 115, Yes, Yes], [[a0, s], 110, 0, 110, 115, Yes, Yes], [[a1, s], 110, 0, 110, 115, Yes, Yes], [[a0, a1, s], 110, 0, 110, 115, Yes, Yes], [[a2, s], 110, 0, 110, 115, Yes, Yes], [[a0, a2, s], 110, 0, 110, 115, Yes, Yes], [[a1, a2, s], 110, 0, 110, 115, Yes, Yes], [[a0, a1, a2, s], 110, 0, 110, 115, Yes, Yes], [[a3, s], 110, 0, 110, 115, Yes, Yes], [[a0, a3, s], 110, 0, 110, 115, Yes, Yes], [[a1, a3, s], 110, 0, 110, 115, Yes, Yes], [[a0, a1, a3, s], 110, 0, 110, 115, Yes, Yes], [[a2, a3, s], 110, 0, 110, 115, Yes, Yes], [[a0, a2, a3, s], 110, 0, 110, 115, Yes, Yes], [[a1, a2, a3, s], 110, 0, 110, 115, Yes, Yes], [[a0, a1, a2, a3, s], 110, 0, 110, 115, Yes, Yes], [[a4, s], 110, 0, 110, 115, Yes, Yes], [[a0, a4, s], 110, 0, 110, 115, Yes, Yes], [[a1, a4, s], 110, 0, 110, 115, Yes, Yes], [[a0, a1, a4, s], 110, 0, 110, 115, Yes, Yes], [[a2, a4, s], 110, 0, 110, 115, Yes, Yes], [[a0, a2, a4, s], 110, 0, 110, 115, Yes, Yes], [[a1, a2, a4, s], 110, 0, 110, 115, Yes, Yes], [[a0, a1, a2, a4, s], 110, 0, 110, 115, Yes, Yes], [[a3, a4, s], 110, 0, 110, 115, Yes, Yes], [[a0, a3, a4, s], 110, 0, 110, 115, Yes, Yes], [[a1, a3, a4, s], 110, 0, 110, 115, Yes, Yes], [[a0, a1, a3, a4, s], 110, 0, 110, 115, Yes, Yes], [[a2, a3, a4, s], 110, 0, 110, 115, Yes, Yes], [[a0, a2, a3, a4, s], 110, 0, 110, 115, Yes, Yes], [[a1, a2, a3, a4, s], 110, 0, 110, 115, Yes, Yes], [[a0, a1, a2, a3, a4, s], 110, 0, 110, 115, Yes, Yes]], Yes, No, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum).
Reviewed step-by-step derivation
Max-flow min-cut theorem · qf-mod46-mfmc-flow-cut-upper-bound · candidate
46. A Feasible Flow Is Bounded above by Every Cut, Not Automatically Maximum
Use complete directed vertices=[s, a0, a1, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t]], source=s, sink=t, aligned nonnegative exact capacities=[22, 22, 23, 23], candidate flows=[21, 21, 22, 22]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (source_value, sink_value, flow_cut_rows, all_cut_bounds_hold, automatic_maximum_claim, certificate).
Cut capacity counts outgoing edges only; net flow subtracts incoming flow. No automatic_maximum_claim does not classify this particular flow as necessarily nonmaximum.
View answer and explanation
(43, 43, [[[s], 43, 0, 43, 45, Yes, Yes], [[a0, s], 43, 0, 43, 45, Yes, Yes], [[a1, s], 43, 0, 43, 45, Yes, Yes], [[a0, a1, s], 43, 0, 43, 45, Yes, Yes]], Yes, No, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum)
Require nonnegative capacity-bounded flows and conservation at every internal vertex. For every cut compute outgoing flow, incoming flow, their difference, capacity, net/source-value equality and the upper bound. The fixed No withholds automatic maximality; it does not assert that this particular feasible flow is nonmaximum. The exact result, retaining the requested order, is (43, 43, [[[s], 43, 0, 43, 45, Yes, Yes], [[a0, s], 43, 0, 43, 45, Yes, Yes], [[a1, s], 43, 0, 43, 45, Yes, Yes], [[a0, a1, s], 43, 0, 43, 45, Yes, Yes]], Yes, No, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum).
Reviewed step-by-step derivation
Max-flow min-cut theorem · qf-mod46-mfmc-flow-cut-upper-bound · candidate
47. A Feasible Flow Is Bounded above by Every Cut, Not Automatically Maximum
Use complete directed vertices=[s, a0, a1, a2, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t], [s, a2], [a2, t], [a0, s]], source=s, sink=t, aligned nonnegative exact capacities=[23, 23, 24, 24, 25, 25, 28], candidate flows=[22, 22, 23, 23, 24, 24, 0]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (source_value, sink_value, flow_cut_rows, all_cut_bounds_hold, automatic_maximum_claim, certificate).
Cut capacity counts outgoing edges only; net flow subtracts incoming flow. No automatic_maximum_claim does not classify this particular flow as necessarily nonmaximum.
View answer and explanation
(69, 69, [[[s], 69, 0, 69, 72, Yes, Yes], [[a0, s], 69, 0, 69, 72, Yes, Yes], [[a1, s], 69, 0, 69, 72, Yes, Yes], [[a0, a1, s], 69, 0, 69, 72, Yes, Yes], [[a2, s], 69, 0, 69, 72, Yes, Yes], [[a0, a2, s], 69, 0, 69, 72, Yes, Yes], [[a1, a2, s], 69, 0, 69, 72, Yes, Yes], [[a0, a1, a2, s], 69, 0, 69, 72, Yes, Yes]], Yes, No, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum)
Require nonnegative capacity-bounded flows and conservation at every internal vertex. For every cut compute outgoing flow, incoming flow, their difference, capacity, net/source-value equality and the upper bound. The fixed No withholds automatic maximality; it does not assert that this particular feasible flow is nonmaximum. The exact result, retaining the requested order, is (69, 69, [[[s], 69, 0, 69, 72, Yes, Yes], [[a0, s], 69, 0, 69, 72, Yes, Yes], [[a1, s], 69, 0, 69, 72, Yes, Yes], [[a0, a1, s], 69, 0, 69, 72, Yes, Yes], [[a2, s], 69, 0, 69, 72, Yes, Yes], [[a0, a2, s], 69, 0, 69, 72, Yes, Yes], [[a1, a2, s], 69, 0, 69, 72, Yes, Yes], [[a0, a1, a2, s], 69, 0, 69, 72, Yes, Yes]], Yes, No, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum).
Reviewed step-by-step derivation
Max-flow min-cut theorem · qf-mod46-mfmc-flow-cut-upper-bound · candidate
48. A Feasible Flow Is Bounded above by Every Cut, Not Automatically Maximum
Use complete directed vertices=[s, a0, a1, a2, a3, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t], [s, a2], [a2, t], [s, a3], [a3, t]], source=s, sink=t, aligned nonnegative exact capacities=[24, 24, 25, 25, 26, 26, 27, 27], candidate flows=[23, 23, 24, 24, 25, 25, 26, 26]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (source_value, sink_value, flow_cut_rows, all_cut_bounds_hold, automatic_maximum_claim, certificate).
Cut capacity counts outgoing edges only; net flow subtracts incoming flow. No automatic_maximum_claim does not classify this particular flow as necessarily nonmaximum.
View answer and explanation
(98, 98, [[[s], 98, 0, 98, 102, Yes, Yes], [[a0, s], 98, 0, 98, 102, Yes, Yes], [[a1, s], 98, 0, 98, 102, Yes, Yes], [[a0, a1, s], 98, 0, 98, 102, Yes, Yes], [[a2, s], 98, 0, 98, 102, Yes, Yes], [[a0, a2, s], 98, 0, 98, 102, Yes, Yes], [[a1, a2, s], 98, 0, 98, 102, Yes, Yes], [[a0, a1, a2, s], 98, 0, 98, 102, Yes, Yes], [[a3, s], 98, 0, 98, 102, Yes, Yes], [[a0, a3, s], 98, 0, 98, 102, Yes, Yes], [[a1, a3, s], 98, 0, 98, 102, Yes, Yes], [[a0, a1, a3, s], 98, 0, 98, 102, Yes, Yes], [[a2, a3, s], 98, 0, 98, 102, Yes, Yes], [[a0, a2, a3, s], 98, 0, 98, 102, Yes, Yes], [[a1, a2, a3, s], 98, 0, 98, 102, Yes, Yes], [[a0, a1, a2, a3, s], 98, 0, 98, 102, Yes, Yes]], Yes, No, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum)
Require nonnegative capacity-bounded flows and conservation at every internal vertex. For every cut compute outgoing flow, incoming flow, their difference, capacity, net/source-value equality and the upper bound. The fixed No withholds automatic maximality; it does not assert that this particular feasible flow is nonmaximum. The exact result, retaining the requested order, is (98, 98, [[[s], 98, 0, 98, 102, Yes, Yes], [[a0, s], 98, 0, 98, 102, Yes, Yes], [[a1, s], 98, 0, 98, 102, Yes, Yes], [[a0, a1, s], 98, 0, 98, 102, Yes, Yes], [[a2, s], 98, 0, 98, 102, Yes, Yes], [[a0, a2, s], 98, 0, 98, 102, Yes, Yes], [[a1, a2, s], 98, 0, 98, 102, Yes, Yes], [[a0, a1, a2, s], 98, 0, 98, 102, Yes, Yes], [[a3, s], 98, 0, 98, 102, Yes, Yes], [[a0, a3, s], 98, 0, 98, 102, Yes, Yes], [[a1, a3, s], 98, 0, 98, 102, Yes, Yes], [[a0, a1, a3, s], 98, 0, 98, 102, Yes, Yes], [[a2, a3, s], 98, 0, 98, 102, Yes, Yes], [[a0, a2, a3, s], 98, 0, 98, 102, Yes, Yes], [[a1, a2, a3, s], 98, 0, 98, 102, Yes, Yes], [[a0, a1, a2, a3, s], 98, 0, 98, 102, Yes, Yes]], Yes, No, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum).
Reviewed step-by-step derivation
Max-flow min-cut theorem · qf-mod46-mfmc-flow-cut-upper-bound · candidate
49. A Feasible Flow Is Bounded above by Every Cut, Not Automatically Maximum
Use complete directed vertices=[s, a0, a1, a2, a3, a4, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t], [s, a2], [a2, t], [s, a3], [a3, t], [s, a4], [a4, t]], source=s, sink=t, aligned nonnegative exact capacities=[25, 25, 26, 26, 27, 27, 28, 28, 29, 29], candidate flows=[24, 24, 25, 25, 26, 26, 27, 27, 28, 28]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (source_value, sink_value, flow_cut_rows, all_cut_bounds_hold, automatic_maximum_claim, certificate).
Cut capacity counts outgoing edges only; net flow subtracts incoming flow. No automatic_maximum_claim does not classify this particular flow as necessarily nonmaximum.
View answer and explanation
(130, 130, [[[s], 130, 0, 130, 135, Yes, Yes], [[a0, s], 130, 0, 130, 135, Yes, Yes], [[a1, s], 130, 0, 130, 135, Yes, Yes], [[a0, a1, s], 130, 0, 130, 135, Yes, Yes], [[a2, s], 130, 0, 130, 135, Yes, Yes], [[a0, a2, s], 130, 0, 130, 135, Yes, Yes], [[a1, a2, s], 130, 0, 130, 135, Yes, Yes], [[a0, a1, a2, s], 130, 0, 130, 135, Yes, Yes], [[a3, s], 130, 0, 130, 135, Yes, Yes], [[a0, a3, s], 130, 0, 130, 135, Yes, Yes], [[a1, a3, s], 130, 0, 130, 135, Yes, Yes], [[a0, a1, a3, s], 130, 0, 130, 135, Yes, Yes], [[a2, a3, s], 130, 0, 130, 135, Yes, Yes], [[a0, a2, a3, s], 130, 0, 130, 135, Yes, Yes], [[a1, a2, a3, s], 130, 0, 130, 135, Yes, Yes], [[a0, a1, a2, a3, s], 130, 0, 130, 135, Yes, Yes], [[a4, s], 130, 0, 130, 135, Yes, Yes], [[a0, a4, s], 130, 0, 130, 135, Yes, Yes], [[a1, a4, s], 130, 0, 130, 135, Yes, Yes], [[a0, a1, a4, s], 130, 0, 130, 135, Yes, Yes], [[a2, a4, s], 130, 0, 130, 135, Yes, Yes], [[a0, a2, a4, s], 130, 0, 130, 135, Yes, Yes], [[a1, a2, a4, s], 130, 0, 130, 135, Yes, Yes], [[a0, a1, a2, a4, s], 130, 0, 130, 135, Yes, Yes], [[a3, a4, s], 130, 0, 130, 135, Yes, Yes], [[a0, a3, a4, s], 130, 0, 130, 135, Yes, Yes], [[a1, a3, a4, s], 130, 0, 130, 135, Yes, Yes], [[a0, a1, a3, a4, s], 130, 0, 130, 135, Yes, Yes], [[a2, a3, a4, s], 130, 0, 130, 135, Yes, Yes], [[a0, a2, a3, a4, s], 130, 0, 130, 135, Yes, Yes], [[a1, a2, a3, a4, s], 130, 0, 130, 135, Yes, Yes], [[a0, a1, a2, a3, a4, s], 130, 0, 130, 135, Yes, Yes]], Yes, No, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum)
Require nonnegative capacity-bounded flows and conservation at every internal vertex. For every cut compute outgoing flow, incoming flow, their difference, capacity, net/source-value equality and the upper bound. The fixed No withholds automatic maximality; it does not assert that this particular feasible flow is nonmaximum. The exact result, retaining the requested order, is (130, 130, [[[s], 130, 0, 130, 135, Yes, Yes], [[a0, s], 130, 0, 130, 135, Yes, Yes], [[a1, s], 130, 0, 130, 135, Yes, Yes], [[a0, a1, s], 130, 0, 130, 135, Yes, Yes], [[a2, s], 130, 0, 130, 135, Yes, Yes], [[a0, a2, s], 130, 0, 130, 135, Yes, Yes], [[a1, a2, s], 130, 0, 130, 135, Yes, Yes], [[a0, a1, a2, s], 130, 0, 130, 135, Yes, Yes], [[a3, s], 130, 0, 130, 135, Yes, Yes], [[a0, a3, s], 130, 0, 130, 135, Yes, Yes], [[a1, a3, s], 130, 0, 130, 135, Yes, Yes], [[a0, a1, a3, s], 130, 0, 130, 135, Yes, Yes], [[a2, a3, s], 130, 0, 130, 135, Yes, Yes], [[a0, a2, a3, s], 130, 0, 130, 135, Yes, Yes], [[a1, a2, a3, s], 130, 0, 130, 135, Yes, Yes], [[a0, a1, a2, a3, s], 130, 0, 130, 135, Yes, Yes], [[a4, s], 130, 0, 130, 135, Yes, Yes], [[a0, a4, s], 130, 0, 130, 135, Yes, Yes], [[a1, a4, s], 130, 0, 130, 135, Yes, Yes], [[a0, a1, a4, s], 130, 0, 130, 135, Yes, Yes], [[a2, a4, s], 130, 0, 130, 135, Yes, Yes], [[a0, a2, a4, s], 130, 0, 130, 135, Yes, Yes], [[a1, a2, a4, s], 130, 0, 130, 135, Yes, Yes], [[a0, a1, a2, a4, s], 130, 0, 130, 135, Yes, Yes], [[a3, a4, s], 130, 0, 130, 135, Yes, Yes], [[a0, a3, a4, s], 130, 0, 130, 135, Yes, Yes], [[a1, a3, a4, s], 130, 0, 130, 135, Yes, Yes], [[a0, a1, a3, a4, s], 130, 0, 130, 135, Yes, Yes], [[a2, a3, a4, s], 130, 0, 130, 135, Yes, Yes], [[a0, a2, a3, a4, s], 130, 0, 130, 135, Yes, Yes], [[a1, a2, a3, a4, s], 130, 0, 130, 135, Yes, Yes], [[a0, a1, a2, a3, a4, s], 130, 0, 130, 135, Yes, Yes]], Yes, No, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum).
Reviewed step-by-step derivation
Max-flow min-cut theorem · qf-mod46-mfmc-flow-cut-upper-bound · candidate
50. A Feasible Flow Is Bounded above by Every Cut, Not Automatically Maximum
Use complete directed vertices=[s, a0, a1, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t], [a0, s]], source=s, sink=t, aligned nonnegative exact capacities=[26, 26, 27, 27, 31], candidate flows=[25, 25, 26, 26, 0]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (source_value, sink_value, flow_cut_rows, all_cut_bounds_hold, automatic_maximum_claim, certificate).
Cut capacity counts outgoing edges only; net flow subtracts incoming flow. No automatic_maximum_claim does not classify this particular flow as necessarily nonmaximum.
View answer and explanation
(51, 51, [[[s], 51, 0, 51, 53, Yes, Yes], [[a0, s], 51, 0, 51, 53, Yes, Yes], [[a1, s], 51, 0, 51, 53, Yes, Yes], [[a0, a1, s], 51, 0, 51, 53, Yes, Yes]], Yes, No, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum)
Require nonnegative capacity-bounded flows and conservation at every internal vertex. For every cut compute outgoing flow, incoming flow, their difference, capacity, net/source-value equality and the upper bound. The fixed No withholds automatic maximality; it does not assert that this particular feasible flow is nonmaximum. The exact result, retaining the requested order, is (51, 51, [[[s], 51, 0, 51, 53, Yes, Yes], [[a0, s], 51, 0, 51, 53, Yes, Yes], [[a1, s], 51, 0, 51, 53, Yes, Yes], [[a0, a1, s], 51, 0, 51, 53, Yes, Yes]], Yes, No, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum).
Reviewed step-by-step derivation
Max-flow min-cut theorem · qf-mod46-mfmc-flow-cut-upper-bound · candidate
51. The Minimum Cut Is Selected from the Complete Cut Family
Use complete directed vertices=[s, a0, a1, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t], [a0, s]], source=s, sink=t, aligned nonnegative exact capacities=[2, 2, 3, 3, 7]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (cut_rows, minimum_capacity, minimum_cuts, certificate).
View answer and explanation
([[[s], [[[s, a0], 2], [[s, a1], 3]], 5], [[a0, s], [[[a0, t], 2], [[s, a1], 3]], 5], [[a1, s], [[[a1, t], 3], [[s, a0], 2]], 5], [[a0, a1, s], [[[a0, t], 2], [[a1, t], 3]], 5]], 5, [[s], [a0, s], [a1, s], [a0, a1, s]], minimum_cut_is_selected_from_the_complete_cut_family)
Retain all cut rows, the least exact capacity and every minimizing source-side set. A single small cut without complete comparison does not establish a minimum. The finite network certificate does not prove the general theorem. The exact result, retaining the requested order, is ([[[s], [[[s, a0], 2], [[s, a1], 3]], 5], [[a0, s], [[[a0, t], 2], [[s, a1], 3]], 5], [[a1, s], [[[a1, t], 3], [[s, a0], 2]], 5], [[a0, a1, s], [[[a0, t], 2], [[a1, t], 3]], 5]], 5, [[s], [a0, s], [a1, s], [a0, a1, s]], minimum_cut_is_selected_from_the_complete_cut_family).
Reviewed step-by-step derivation
Max-flow min-cut theorem · qf-mod46-mfmc-minimum-cut-audit · candidate
52. The Minimum Cut Is Selected from the Complete Cut Family
Use complete directed vertices=[s, a0, a1, a2, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t], [s, a2], [a2, t]], source=s, sink=t, aligned nonnegative exact capacities=[3, 3, 4, 4, 5, 5]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (cut_rows, minimum_capacity, minimum_cuts, certificate).
View answer and explanation
([[[s], [[[s, a0], 3], [[s, a1], 4], [[s, a2], 5]], 12], [[a0, s], [[[a0, t], 3], [[s, a1], 4], [[s, a2], 5]], 12], [[a1, s], [[[a1, t], 4], [[s, a0], 3], [[s, a2], 5]], 12], [[a0, a1, s], [[[a0, t], 3], [[a1, t], 4], [[s, a2], 5]], 12], [[a2, s], [[[a2, t], 5], [[s, a0], 3], [[s, a1], 4]], 12], [[a0, a2, s], [[[a0, t], 3], [[a2, t], 5], [[s, a1], 4]], 12], [[a1, a2, s], [[[a1, t], 4], [[a2, t], 5], [[s, a0], 3]], 12], [[a0, a1, a2, s], [[[a0, t], 3], [[a1, t], 4], [[a2, t], 5]], 12]], 12, [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s]], minimum_cut_is_selected_from_the_complete_cut_family)
Retain all cut rows, the least exact capacity and every minimizing source-side set. A single small cut without complete comparison does not establish a minimum. The finite network certificate does not prove the general theorem. The exact result, retaining the requested order, is ([[[s], [[[s, a0], 3], [[s, a1], 4], [[s, a2], 5]], 12], [[a0, s], [[[a0, t], 3], [[s, a1], 4], [[s, a2], 5]], 12], [[a1, s], [[[a1, t], 4], [[s, a0], 3], [[s, a2], 5]], 12], [[a0, a1, s], [[[a0, t], 3], [[a1, t], 4], [[s, a2], 5]], 12], [[a2, s], [[[a2, t], 5], [[s, a0], 3], [[s, a1], 4]], 12], [[a0, a2, s], [[[a0, t], 3], [[a2, t], 5], [[s, a1], 4]], 12], [[a1, a2, s], [[[a1, t], 4], [[a2, t], 5], [[s, a0], 3]], 12], [[a0, a1, a2, s], [[[a0, t], 3], [[a1, t], 4], [[a2, t], 5]], 12]], 12, [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s]], minimum_cut_is_selected_from_the_complete_cut_family).
Reviewed step-by-step derivation
Max-flow min-cut theorem · qf-mod46-mfmc-minimum-cut-audit · candidate
53. The Minimum Cut Is Selected from the Complete Cut Family
Use complete directed vertices=[s, a0, a1, a2, a3, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t], [s, a2], [a2, t], [s, a3], [a3, t]], source=s, sink=t, aligned nonnegative exact capacities=[4, 4, 5, 5, 6, 6, 7, 7]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (cut_rows, minimum_capacity, minimum_cuts, certificate).
View answer and explanation
([[[s], [[[s, a0], 4], [[s, a1], 5], [[s, a2], 6], [[s, a3], 7]], 22], [[a0, s], [[[a0, t], 4], [[s, a1], 5], [[s, a2], 6], [[s, a3], 7]], 22], [[a1, s], [[[a1, t], 5], [[s, a0], 4], [[s, a2], 6], [[s, a3], 7]], 22], [[a0, a1, s], [[[a0, t], 4], [[a1, t], 5], [[s, a2], 6], [[s, a3], 7]], 22], [[a2, s], [[[a2, t], 6], [[s, a0], 4], [[s, a1], 5], [[s, a3], 7]], 22], [[a0, a2, s], [[[a0, t], 4], [[a2, t], 6], [[s, a1], 5], [[s, a3], 7]], 22], [[a1, a2, s], [[[a1, t], 5], [[a2, t], 6], [[s, a0], 4], [[s, a3], 7]], 22], [[a0, a1, a2, s], [[[a0, t], 4], [[a1, t], 5], [[a2, t], 6], [[s, a3], 7]], 22], [[a3, s], [[[a3, t], 7], [[s, a0], 4], [[s, a1], 5], [[s, a2], 6]], 22], [[a0, a3, s], [[[a0, t], 4], [[a3, t], 7], [[s, a1], 5], [[s, a2], 6]], 22], [[a1, a3, s], [[[a1, t], 5], [[a3, t], 7], [[s, a0], 4], [[s, a2], 6]], 22], [[a0, a1, a3, s], [[[a0, t], 4], [[a1, t], 5], [[a3, t], 7], [[s, a2], 6]], 22], [[a2, a3, s], [[[a2, t], 6], [[a3, t], 7], [[s, a0], 4], [[s, a1], 5]], 22], [[a0, a2, a3, s], [[[a0, t], 4], [[a2, t], 6], [[a3, t], 7], [[s, a1], 5]], 22], [[a1, a2, a3, s], [[[a1, t], 5], [[a2, t], 6], [[a3, t], 7], [[s, a0], 4]], 22], [[a0, a1, a2, a3, s], [[[a0, t], 4], [[a1, t], 5], [[a2, t], 6], [[a3, t], 7]], 22]], 22, [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s], [a3, s], [a0, a3, s], [a1, a3, s], [a0, a1, a3, s], [a2, a3, s], [a0, a2, a3, s], [a1, a2, a3, s], [a0, a1, a2, a3, s]], minimum_cut_is_selected_from_the_complete_cut_family)
Retain all cut rows, the least exact capacity and every minimizing source-side set. A single small cut without complete comparison does not establish a minimum. The finite network certificate does not prove the general theorem. The exact result, retaining the requested order, is ([[[s], [[[s, a0], 4], [[s, a1], 5], [[s, a2], 6], [[s, a3], 7]], 22], [[a0, s], [[[a0, t], 4], [[s, a1], 5], [[s, a2], 6], [[s, a3], 7]], 22], [[a1, s], [[[a1, t], 5], [[s, a0], 4], [[s, a2], 6], [[s, a3], 7]], 22], [[a0, a1, s], [[[a0, t], 4], [[a1, t], 5], [[s, a2], 6], [[s, a3], 7]], 22], [[a2, s], [[[a2, t], 6], [[s, a0], 4], [[s, a1], 5], [[s, a3], 7]], 22], [[a0, a2, s], [[[a0, t], 4], [[a2, t], 6], [[s, a1], 5], [[s, a3], 7]], 22], [[a1, a2, s], [[[a1, t], 5], [[a2, t], 6], [[s, a0], 4], [[s, a3], 7]], 22], [[a0, a1, a2, s], [[[a0, t], 4], [[a1, t], 5], [[a2, t], 6], [[s, a3], 7]], 22], [[a3, s], [[[a3, t], 7], [[s, a0], 4], [[s, a1], 5], [[s, a2], 6]], 22], [[a0, a3, s], [[[a0, t], 4], [[a3, t], 7], [[s, a1], 5], [[s, a2], 6]], 22], [[a1, a3, s], [[[a1, t], 5], [[a3, t], 7], [[s, a0], 4], [[s, a2], 6]], 22], [[a0, a1, a3, s], [[[a0, t], 4], [[a1, t], 5], [[a3, t], 7], [[s, a2], 6]], 22], [[a2, a3, s], [[[a2, t], 6], [[a3, t], 7], [[s, a0], 4], [[s, a1], 5]], 22], [[a0, a2, a3, s], [[[a0, t], 4], [[a2, t], 6], [[a3, t], 7], [[s, a1], 5]], 22], [[a1, a2, a3, s], [[[a1, t], 5], [[a2, t], 6], [[a3, t], 7], [[s, a0], 4]], 22], [[a0, a1, a2, a3, s], [[[a0, t], 4], [[a1, t], 5], [[a2, t], 6], [[a3, t], 7]], 22]], 22, [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s], [a3, s], [a0, a3, s], [a1, a3, s], [a0, a1, a3, s], [a2, a3, s], [a0, a2, a3, s], [a1, a2, a3, s], [a0, a1, a2, a3, s]], minimum_cut_is_selected_from_the_complete_cut_family).
Reviewed step-by-step derivation
Max-flow min-cut theorem · qf-mod46-mfmc-minimum-cut-audit · candidate
54. The Minimum Cut Is Selected from the Complete Cut Family
Use complete directed vertices=[s, a0, a1, a2, a3, a4, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t], [s, a2], [a2, t], [s, a3], [a3, t], [s, a4], [a4, t], [a0, s]], source=s, sink=t, aligned nonnegative exact capacities=[5, 5, 6, 6, 7, 7, 8, 8, 9, 9, 10]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (cut_rows, minimum_capacity, minimum_cuts, certificate).
View answer and explanation
([[[s], [[[s, a0], 5], [[s, a1], 6], [[s, a2], 7], [[s, a3], 8], [[s, a4], 9]], 35], [[a0, s], [[[a0, t], 5], [[s, a1], 6], [[s, a2], 7], [[s, a3], 8], [[s, a4], 9]], 35], [[a1, s], [[[a1, t], 6], [[s, a0], 5], [[s, a2], 7], [[s, a3], 8], [[s, a4], 9]], 35], [[a0, a1, s], [[[a0, t], 5], [[a1, t], 6], [[s, a2], 7], [[s, a3], 8], [[s, a4], 9]], 35], [[a2, s], [[[a2, t], 7], [[s, a0], 5], [[s, a1], 6], [[s, a3], 8], [[s, a4], 9]], 35], [[a0, a2, s], [[[a0, t], 5], [[a2, t], 7], [[s, a1], 6], [[s, a3], 8], [[s, a4], 9]], 35], [[a1, a2, s], [[[a1, t], 6], [[a2, t], 7], [[s, a0], 5], [[s, a3], 8], [[s, a4], 9]], 35], [[a0, a1, a2, s], [[[a0, t], 5], [[a1, t], 6], [[a2, t], 7], [[s, a3], 8], [[s, a4], 9]], 35], [[a3, s], [[[a3, t], 8], [[s, a0], 5], [[s, a1], 6], [[s, a2], 7], [[s, a4], 9]], 35], [[a0, a3, s], [[[a0, t], 5], [[a3, t], 8], [[s, a1], 6], [[s, a2], 7], [[s, a4], 9]], 35], [[a1, a3, s], [[[a1, t], 6], [[a3, t], 8], [[s, a0], 5], [[s, a2], 7], [[s, a4], 9]], 35], [[a0, a1, a3, s], [[[a0, t], 5], [[a1, t], 6], [[a3, t], 8], [[s, a2], 7], [[s, a4], 9]], 35], [[a2, a3, s], [[[a2, t], 7], [[a3, t], 8], [[s, a0], 5], [[s, a1], 6], [[s, a4], 9]], 35], [[a0, a2, a3, s], [[[a0, t], 5], [[a2, t], 7], [[a3, t], 8], [[s, a1], 6], [[s, a4], 9]], 35], [[a1, a2, a3, s], [[[a1, t], 6], [[a2, t], 7], [[a3, t], 8], [[s, a0], 5], [[s, a4], 9]], 35], [[a0, a1, a2, a3, s], [[[a0, t], 5], [[a1, t], 6], [[a2, t], 7], [[a3, t], 8], [[s, a4], 9]], 35], [[a4, s], [[[a4, t], 9], [[s, a0], 5], [[s, a1], 6], [[s, a2], 7], [[s, a3], 8]], 35], [[a0, a4, s], [[[a0, t], 5], [[a4, t], 9], [[s, a1], 6], [[s, a2], 7], [[s, a3], 8]], 35], [[a1, a4, s], [[[a1, t], 6], [[a4, t], 9], [[s, a0], 5], [[s, a2], 7], [[s, a3], 8]], 35], [[a0, a1, a4, s], [[[a0, t], 5], [[a1, t], 6], [[a4, t], 9], [[s, a2], 7], [[s, a3], 8]], 35], [[a2, a4, s], [[[a2, t], 7], [[a4, t], 9], [[s, a0], 5], [[s, a1], 6], [[s, a3], 8]], 35], [[a0, a2, a4, s], [[[a0, t], 5], [[a2, t], 7], [[a4, t], 9], [[s, a1], 6], [[s, a3], 8]], 35], [[a1, a2, a4, s], [[[a1, t], 6], [[a2, t], 7], [[a4, t], 9], [[s, a0], 5], [[s, a3], 8]], 35], [[a0, a1, a2, a4, s], [[[a0, t], 5], [[a1, t], 6], [[a2, t], 7], [[a4, t], 9], [[s, a3], 8]], 35], [[a3, a4, s], [[[a3, t], 8], [[a4, t], 9], [[s, a0], 5], [[s, a1], 6], [[s, a2], 7]], 35], [[a0, a3, a4, s], [[[a0, t], 5], [[a3, t], 8], [[a4, t], 9], [[s, a1], 6], [[s, a2], 7]], 35], [[a1, a3, a4, s], [[[a1, t], 6], [[a3, t], 8], [[a4, t], 9], [[s, a0], 5], [[s, a2], 7]], 35], [[a0, a1, a3, a4, s], [[[a0, t], 5], [[a1, t], 6], [[a3, t], 8], [[a4, t], 9], [[s, a2], 7]], 35], [[a2, a3, a4, s], [[[a2, t], 7], [[a3, t], 8], [[a4, t], 9], [[s, a0], 5], [[s, a1], 6]], 35], [[a0, a2, a3, a4, s], [[[a0, t], 5], [[a2, t], 7], [[a3, t], 8], [[a4, t], 9], [[s, a1], 6]], 35], [[a1, a2, a3, a4, s], [[[a1, t], 6], [[a2, t], 7], [[a3, t], 8], [[a4, t], 9], [[s, a0], 5]], 35], [[a0, a1, a2, a3, a4, s], [[[a0, t], 5], [[a1, t], 6], [[a2, t], 7], [[a3, t], 8], [[a4, t], 9]], 35]], 35, [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s], [a3, s], [a0, a3, s], [a1, a3, s], [a0, a1, a3, s], [a2, a3, s], [a0, a2, a3, s], [a1, a2, a3, s], [a0, a1, a2, a3, s], [a4, s], [a0, a4, s], [a1, a4, s], [a0, a1, a4, s], [a2, a4, s], [a0, a2, a4, s], [a1, a2, a4, s], [a0, a1, a2, a4, s], [a3, a4, s], [a0, a3, a4, s], [a1, a3, a4, s], [a0, a1, a3, a4, s], [a2, a3, a4, s], [a0, a2, a3, a4, s], [a1, a2, a3, a4, s], [a0, a1, a2, a3, a4, s]], minimum_cut_is_selected_from_the_complete_cut_family)
Retain all cut rows, the least exact capacity and every minimizing source-side set. A single small cut without complete comparison does not establish a minimum. The finite network certificate does not prove the general theorem. The exact result, retaining the requested order, is ([[[s], [[[s, a0], 5], [[s, a1], 6], [[s, a2], 7], [[s, a3], 8], [[s, a4], 9]], 35], [[a0, s], [[[a0, t], 5], [[s, a1], 6], [[s, a2], 7], [[s, a3], 8], [[s, a4], 9]], 35], [[a1, s], [[[a1, t], 6], [[s, a0], 5], [[s, a2], 7], [[s, a3], 8], [[s, a4], 9]], 35], [[a0, a1, s], [[[a0, t], 5], [[a1, t], 6], [[s, a2], 7], [[s, a3], 8], [[s, a4], 9]], 35], [[a2, s], [[[a2, t], 7], [[s, a0], 5], [[s, a1], 6], [[s, a3], 8], [[s, a4], 9]], 35], [[a0, a2, s], [[[a0, t], 5], [[a2, t], 7], [[s, a1], 6], [[s, a3], 8], [[s, a4], 9]], 35], [[a1, a2, s], [[[a1, t], 6], [[a2, t], 7], [[s, a0], 5], [[s, a3], 8], [[s, a4], 9]], 35], [[a0, a1, a2, s], [[[a0, t], 5], [[a1, t], 6], [[a2, t], 7], [[s, a3], 8], [[s, a4], 9]], 35], [[a3, s], [[[a3, t], 8], [[s, a0], 5], [[s, a1], 6], [[s, a2], 7], [[s, a4], 9]], 35], [[a0, a3, s], [[[a0, t], 5], [[a3, t], 8], [[s, a1], 6], [[s, a2], 7], [[s, a4], 9]], 35], [[a1, a3, s], [[[a1, t], 6], [[a3, t], 8], [[s, a0], 5], [[s, a2], 7], [[s, a4], 9]], 35], [[a0, a1, a3, s], [[[a0, t], 5], [[a1, t], 6], [[a3, t], 8], [[s, a2], 7], [[s, a4], 9]], 35], [[a2, a3, s], [[[a2, t], 7], [[a3, t], 8], [[s, a0], 5], [[s, a1], 6], [[s, a4], 9]], 35], [[a0, a2, a3, s], [[[a0, t], 5], [[a2, t], 7], [[a3, t], 8], [[s, a1], 6], [[s, a4], 9]], 35], [[a1, a2, a3, s], [[[a1, t], 6], [[a2, t], 7], [[a3, t], 8], [[s, a0], 5], [[s, a4], 9]], 35], [[a0, a1, a2, a3, s], [[[a0, t], 5], [[a1, t], 6], [[a2, t], 7], [[a3, t], 8], [[s, a4], 9]], 35], [[a4, s], [[[a4, t], 9], [[s, a0], 5], [[s, a1], 6], [[s, a2], 7], [[s, a3], 8]], 35], [[a0, a4, s], [[[a0, t], 5], [[a4, t], 9], [[s, a1], 6], [[s, a2], 7], [[s, a3], 8]], 35], [[a1, a4, s], [[[a1, t], 6], [[a4, t], 9], [[s, a0], 5], [[s, a2], 7], [[s, a3], 8]], 35], [[a0, a1, a4, s], [[[a0, t], 5], [[a1, t], 6], [[a4, t], 9], [[s, a2], 7], [[s, a3], 8]], 35], [[a2, a4, s], [[[a2, t], 7], [[a4, t], 9], [[s, a0], 5], [[s, a1], 6], [[s, a3], 8]], 35], [[a0, a2, a4, s], [[[a0, t], 5], [[a2, t], 7], [[a4, t], 9], [[s, a1], 6], [[s, a3], 8]], 35], [[a1, a2, a4, s], [[[a1, t], 6], [[a2, t], 7], [[a4, t], 9], [[s, a0], 5], [[s, a3], 8]], 35], [[a0, a1, a2, a4, s], [[[a0, t], 5], [[a1, t], 6], [[a2, t], 7], [[a4, t], 9], [[s, a3], 8]], 35], [[a3, a4, s], [[[a3, t], 8], [[a4, t], 9], [[s, a0], 5], [[s, a1], 6], [[s, a2], 7]], 35], [[a0, a3, a4, s], [[[a0, t], 5], [[a3, t], 8], [[a4, t], 9], [[s, a1], 6], [[s, a2], 7]], 35], [[a1, a3, a4, s], [[[a1, t], 6], [[a3, t], 8], [[a4, t], 9], [[s, a0], 5], [[s, a2], 7]], 35], [[a0, a1, a3, a4, s], [[[a0, t], 5], [[a1, t], 6], [[a3, t], 8], [[a4, t], 9], [[s, a2], 7]], 35], [[a2, a3, a4, s], [[[a2, t], 7], [[a3, t], 8], [[a4, t], 9], [[s, a0], 5], [[s, a1], 6]], 35], [[a0, a2, a3, a4, s], [[[a0, t], 5], [[a2, t], 7], [[a3, t], 8], [[a4, t], 9], [[s, a1], 6]], 35], [[a1, a2, a3, a4, s], [[[a1, t], 6], [[a2, t], 7], [[a3, t], 8], [[a4, t], 9], [[s, a0], 5]], 35], [[a0, a1, a2, a3, a4, s], [[[a0, t], 5], [[a1, t], 6], [[a2, t], 7], [[a3, t], 8], [[a4, t], 9]], 35]], 35, [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s], [a3, s], [a0, a3, s], [a1, a3, s], [a0, a1, a3, s], [a2, a3, s], [a0, a2, a3, s], [a1, a2, a3, s], [a0, a1, a2, a3, s], [a4, s], [a0, a4, s], [a1, a4, s], [a0, a1, a4, s], [a2, a4, s], [a0, a2, a4, s], [a1, a2, a4, s], [a0, a1, a2, a4, s], [a3, a4, s], [a0, a3, a4, s], [a1, a3, a4, s], [a0, a1, a3, a4, s], [a2, a3, a4, s], [a0, a2, a3, a4, s], [a1, a2, a3, a4, s], [a0, a1, a2, a3, a4, s]], minimum_cut_is_selected_from_the_complete_cut_family).
Reviewed step-by-step derivation
Max-flow min-cut theorem · qf-mod46-mfmc-minimum-cut-audit · candidate
55. The Minimum Cut Is Selected from the Complete Cut Family
Use complete directed vertices=[s, a0, a1, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t]], source=s, sink=t, aligned nonnegative exact capacities=[6, 6, 7, 7]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (cut_rows, minimum_capacity, minimum_cuts, certificate).
View answer and explanation
([[[s], [[[s, a0], 6], [[s, a1], 7]], 13], [[a0, s], [[[a0, t], 6], [[s, a1], 7]], 13], [[a1, s], [[[a1, t], 7], [[s, a0], 6]], 13], [[a0, a1, s], [[[a0, t], 6], [[a1, t], 7]], 13]], 13, [[s], [a0, s], [a1, s], [a0, a1, s]], minimum_cut_is_selected_from_the_complete_cut_family)
Retain all cut rows, the least exact capacity and every minimizing source-side set. A single small cut without complete comparison does not establish a minimum. The finite network certificate does not prove the general theorem. The exact result, retaining the requested order, is ([[[s], [[[s, a0], 6], [[s, a1], 7]], 13], [[a0, s], [[[a0, t], 6], [[s, a1], 7]], 13], [[a1, s], [[[a1, t], 7], [[s, a0], 6]], 13], [[a0, a1, s], [[[a0, t], 6], [[a1, t], 7]], 13]], 13, [[s], [a0, s], [a1, s], [a0, a1, s]], minimum_cut_is_selected_from_the_complete_cut_family).
Reviewed step-by-step derivation
Max-flow min-cut theorem · qf-mod46-mfmc-minimum-cut-audit · candidate
56. The Minimum Cut Is Selected from the Complete Cut Family
Use complete directed vertices=[s, a0, a1, a2, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t], [s, a2], [a2, t]], source=s, sink=t, aligned nonnegative exact capacities=[7, 7, 8, 8, 9, 9]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (cut_rows, minimum_capacity, minimum_cuts, certificate).
View answer and explanation
([[[s], [[[s, a0], 7], [[s, a1], 8], [[s, a2], 9]], 24], [[a0, s], [[[a0, t], 7], [[s, a1], 8], [[s, a2], 9]], 24], [[a1, s], [[[a1, t], 8], [[s, a0], 7], [[s, a2], 9]], 24], [[a0, a1, s], [[[a0, t], 7], [[a1, t], 8], [[s, a2], 9]], 24], [[a2, s], [[[a2, t], 9], [[s, a0], 7], [[s, a1], 8]], 24], [[a0, a2, s], [[[a0, t], 7], [[a2, t], 9], [[s, a1], 8]], 24], [[a1, a2, s], [[[a1, t], 8], [[a2, t], 9], [[s, a0], 7]], 24], [[a0, a1, a2, s], [[[a0, t], 7], [[a1, t], 8], [[a2, t], 9]], 24]], 24, [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s]], minimum_cut_is_selected_from_the_complete_cut_family)
Retain all cut rows, the least exact capacity and every minimizing source-side set. A single small cut without complete comparison does not establish a minimum. The finite network certificate does not prove the general theorem. The exact result, retaining the requested order, is ([[[s], [[[s, a0], 7], [[s, a1], 8], [[s, a2], 9]], 24], [[a0, s], [[[a0, t], 7], [[s, a1], 8], [[s, a2], 9]], 24], [[a1, s], [[[a1, t], 8], [[s, a0], 7], [[s, a2], 9]], 24], [[a0, a1, s], [[[a0, t], 7], [[a1, t], 8], [[s, a2], 9]], 24], [[a2, s], [[[a2, t], 9], [[s, a0], 7], [[s, a1], 8]], 24], [[a0, a2, s], [[[a0, t], 7], [[a2, t], 9], [[s, a1], 8]], 24], [[a1, a2, s], [[[a1, t], 8], [[a2, t], 9], [[s, a0], 7]], 24], [[a0, a1, a2, s], [[[a0, t], 7], [[a1, t], 8], [[a2, t], 9]], 24]], 24, [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s]], minimum_cut_is_selected_from_the_complete_cut_family).
Reviewed step-by-step derivation
Max-flow min-cut theorem · qf-mod46-mfmc-minimum-cut-audit · candidate
57. The Minimum Cut Is Selected from the Complete Cut Family
Use complete directed vertices=[s, a0, a1, a2, a3, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t], [s, a2], [a2, t], [s, a3], [a3, t], [a0, s]], source=s, sink=t, aligned nonnegative exact capacities=[8, 8, 9, 9, 10, 10, 11, 11, 13]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (cut_rows, minimum_capacity, minimum_cuts, certificate).
View answer and explanation
([[[s], [[[s, a0], 8], [[s, a1], 9], [[s, a2], 10], [[s, a3], 11]], 38], [[a0, s], [[[a0, t], 8], [[s, a1], 9], [[s, a2], 10], [[s, a3], 11]], 38], [[a1, s], [[[a1, t], 9], [[s, a0], 8], [[s, a2], 10], [[s, a3], 11]], 38], [[a0, a1, s], [[[a0, t], 8], [[a1, t], 9], [[s, a2], 10], [[s, a3], 11]], 38], [[a2, s], [[[a2, t], 10], [[s, a0], 8], [[s, a1], 9], [[s, a3], 11]], 38], [[a0, a2, s], [[[a0, t], 8], [[a2, t], 10], [[s, a1], 9], [[s, a3], 11]], 38], [[a1, a2, s], [[[a1, t], 9], [[a2, t], 10], [[s, a0], 8], [[s, a3], 11]], 38], [[a0, a1, a2, s], [[[a0, t], 8], [[a1, t], 9], [[a2, t], 10], [[s, a3], 11]], 38], [[a3, s], [[[a3, t], 11], [[s, a0], 8], [[s, a1], 9], [[s, a2], 10]], 38], [[a0, a3, s], [[[a0, t], 8], [[a3, t], 11], [[s, a1], 9], [[s, a2], 10]], 38], [[a1, a3, s], [[[a1, t], 9], [[a3, t], 11], [[s, a0], 8], [[s, a2], 10]], 38], [[a0, a1, a3, s], [[[a0, t], 8], [[a1, t], 9], [[a3, t], 11], [[s, a2], 10]], 38], [[a2, a3, s], [[[a2, t], 10], [[a3, t], 11], [[s, a0], 8], [[s, a1], 9]], 38], [[a0, a2, a3, s], [[[a0, t], 8], [[a2, t], 10], [[a3, t], 11], [[s, a1], 9]], 38], [[a1, a2, a3, s], [[[a1, t], 9], [[a2, t], 10], [[a3, t], 11], [[s, a0], 8]], 38], [[a0, a1, a2, a3, s], [[[a0, t], 8], [[a1, t], 9], [[a2, t], 10], [[a3, t], 11]], 38]], 38, [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s], [a3, s], [a0, a3, s], [a1, a3, s], [a0, a1, a3, s], [a2, a3, s], [a0, a2, a3, s], [a1, a2, a3, s], [a0, a1, a2, a3, s]], minimum_cut_is_selected_from_the_complete_cut_family)
Retain all cut rows, the least exact capacity and every minimizing source-side set. A single small cut without complete comparison does not establish a minimum. The finite network certificate does not prove the general theorem. The exact result, retaining the requested order, is ([[[s], [[[s, a0], 8], [[s, a1], 9], [[s, a2], 10], [[s, a3], 11]], 38], [[a0, s], [[[a0, t], 8], [[s, a1], 9], [[s, a2], 10], [[s, a3], 11]], 38], [[a1, s], [[[a1, t], 9], [[s, a0], 8], [[s, a2], 10], [[s, a3], 11]], 38], [[a0, a1, s], [[[a0, t], 8], [[a1, t], 9], [[s, a2], 10], [[s, a3], 11]], 38], [[a2, s], [[[a2, t], 10], [[s, a0], 8], [[s, a1], 9], [[s, a3], 11]], 38], [[a0, a2, s], [[[a0, t], 8], [[a2, t], 10], [[s, a1], 9], [[s, a3], 11]], 38], [[a1, a2, s], [[[a1, t], 9], [[a2, t], 10], [[s, a0], 8], [[s, a3], 11]], 38], [[a0, a1, a2, s], [[[a0, t], 8], [[a1, t], 9], [[a2, t], 10], [[s, a3], 11]], 38], [[a3, s], [[[a3, t], 11], [[s, a0], 8], [[s, a1], 9], [[s, a2], 10]], 38], [[a0, a3, s], [[[a0, t], 8], [[a3, t], 11], [[s, a1], 9], [[s, a2], 10]], 38], [[a1, a3, s], [[[a1, t], 9], [[a3, t], 11], [[s, a0], 8], [[s, a2], 10]], 38], [[a0, a1, a3, s], [[[a0, t], 8], [[a1, t], 9], [[a3, t], 11], [[s, a2], 10]], 38], [[a2, a3, s], [[[a2, t], 10], [[a3, t], 11], [[s, a0], 8], [[s, a1], 9]], 38], [[a0, a2, a3, s], [[[a0, t], 8], [[a2, t], 10], [[a3, t], 11], [[s, a1], 9]], 38], [[a1, a2, a3, s], [[[a1, t], 9], [[a2, t], 10], [[a3, t], 11], [[s, a0], 8]], 38], [[a0, a1, a2, a3, s], [[[a0, t], 8], [[a1, t], 9], [[a2, t], 10], [[a3, t], 11]], 38]], 38, [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s], [a3, s], [a0, a3, s], [a1, a3, s], [a0, a1, a3, s], [a2, a3, s], [a0, a2, a3, s], [a1, a2, a3, s], [a0, a1, a2, a3, s]], minimum_cut_is_selected_from_the_complete_cut_family).
Reviewed step-by-step derivation
Max-flow min-cut theorem · qf-mod46-mfmc-minimum-cut-audit · candidate
58. The Minimum Cut Is Selected from the Complete Cut Family
Use complete directed vertices=[s, a0, a1, a2, a3, a4, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t], [s, a2], [a2, t], [s, a3], [a3, t], [s, a4], [a4, t]], source=s, sink=t, aligned nonnegative exact capacities=[9, 9, 10, 10, 11, 11, 12, 12, 13, 13]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (cut_rows, minimum_capacity, minimum_cuts, certificate).
View answer and explanation
([[[s], [[[s, a0], 9], [[s, a1], 10], [[s, a2], 11], [[s, a3], 12], [[s, a4], 13]], 55], [[a0, s], [[[a0, t], 9], [[s, a1], 10], [[s, a2], 11], [[s, a3], 12], [[s, a4], 13]], 55], [[a1, s], [[[a1, t], 10], [[s, a0], 9], [[s, a2], 11], [[s, a3], 12], [[s, a4], 13]], 55], [[a0, a1, s], [[[a0, t], 9], [[a1, t], 10], [[s, a2], 11], [[s, a3], 12], [[s, a4], 13]], 55], [[a2, s], [[[a2, t], 11], [[s, a0], 9], [[s, a1], 10], [[s, a3], 12], [[s, a4], 13]], 55], [[a0, a2, s], [[[a0, t], 9], [[a2, t], 11], [[s, a1], 10], [[s, a3], 12], [[s, a4], 13]], 55], [[a1, a2, s], [[[a1, t], 10], [[a2, t], 11], [[s, a0], 9], [[s, a3], 12], [[s, a4], 13]], 55], [[a0, a1, a2, s], [[[a0, t], 9], [[a1, t], 10], [[a2, t], 11], [[s, a3], 12], [[s, a4], 13]], 55], [[a3, s], [[[a3, t], 12], [[s, a0], 9], [[s, a1], 10], [[s, a2], 11], [[s, a4], 13]], 55], [[a0, a3, s], [[[a0, t], 9], [[a3, t], 12], [[s, a1], 10], [[s, a2], 11], [[s, a4], 13]], 55], [[a1, a3, s], [[[a1, t], 10], [[a3, t], 12], [[s, a0], 9], [[s, a2], 11], [[s, a4], 13]], 55], [[a0, a1, a3, s], [[[a0, t], 9], [[a1, t], 10], [[a3, t], 12], [[s, a2], 11], [[s, a4], 13]], 55], [[a2, a3, s], [[[a2, t], 11], [[a3, t], 12], [[s, a0], 9], [[s, a1], 10], [[s, a4], 13]], 55], [[a0, a2, a3, s], [[[a0, t], 9], [[a2, t], 11], [[a3, t], 12], [[s, a1], 10], [[s, a4], 13]], 55], [[a1, a2, a3, s], [[[a1, t], 10], [[a2, t], 11], [[a3, t], 12], [[s, a0], 9], [[s, a4], 13]], 55], [[a0, a1, a2, a3, s], [[[a0, t], 9], [[a1, t], 10], [[a2, t], 11], [[a3, t], 12], [[s, a4], 13]], 55], [[a4, s], [[[a4, t], 13], [[s, a0], 9], [[s, a1], 10], [[s, a2], 11], [[s, a3], 12]], 55], [[a0, a4, s], [[[a0, t], 9], [[a4, t], 13], [[s, a1], 10], [[s, a2], 11], [[s, a3], 12]], 55], [[a1, a4, s], [[[a1, t], 10], [[a4, t], 13], [[s, a0], 9], [[s, a2], 11], [[s, a3], 12]], 55], [[a0, a1, a4, s], [[[a0, t], 9], [[a1, t], 10], [[a4, t], 13], [[s, a2], 11], [[s, a3], 12]], 55], [[a2, a4, s], [[[a2, t], 11], [[a4, t], 13], [[s, a0], 9], [[s, a1], 10], [[s, a3], 12]], 55], [[a0, a2, a4, s], [[[a0, t], 9], [[a2, t], 11], [[a4, t], 13], [[s, a1], 10], [[s, a3], 12]], 55], [[a1, a2, a4, s], [[[a1, t], 10], [[a2, t], 11], [[a4, t], 13], [[s, a0], 9], [[s, a3], 12]], 55], [[a0, a1, a2, a4, s], [[[a0, t], 9], [[a1, t], 10], [[a2, t], 11], [[a4, t], 13], [[s, a3], 12]], 55], [[a3, a4, s], [[[a3, t], 12], [[a4, t], 13], [[s, a0], 9], [[s, a1], 10], [[s, a2], 11]], 55], [[a0, a3, a4, s], [[[a0, t], 9], [[a3, t], 12], [[a4, t], 13], [[s, a1], 10], [[s, a2], 11]], 55], [[a1, a3, a4, s], [[[a1, t], 10], [[a3, t], 12], [[a4, t], 13], [[s, a0], 9], [[s, a2], 11]], 55], [[a0, a1, a3, a4, s], [[[a0, t], 9], [[a1, t], 10], [[a3, t], 12], [[a4, t], 13], [[s, a2], 11]], 55], [[a2, a3, a4, s], [[[a2, t], 11], [[a3, t], 12], [[a4, t], 13], [[s, a0], 9], [[s, a1], 10]], 55], [[a0, a2, a3, a4, s], [[[a0, t], 9], [[a2, t], 11], [[a3, t], 12], [[a4, t], 13], [[s, a1], 10]], 55], [[a1, a2, a3, a4, s], [[[a1, t], 10], [[a2, t], 11], [[a3, t], 12], [[a4, t], 13], [[s, a0], 9]], 55], [[a0, a1, a2, a3, a4, s], [[[a0, t], 9], [[a1, t], 10], [[a2, t], 11], [[a3, t], 12], [[a4, t], 13]], 55]], 55, [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s], [a3, s], [a0, a3, s], [a1, a3, s], [a0, a1, a3, s], [a2, a3, s], [a0, a2, a3, s], [a1, a2, a3, s], [a0, a1, a2, a3, s], [a4, s], [a0, a4, s], [a1, a4, s], [a0, a1, a4, s], [a2, a4, s], [a0, a2, a4, s], [a1, a2, a4, s], [a0, a1, a2, a4, s], [a3, a4, s], [a0, a3, a4, s], [a1, a3, a4, s], [a0, a1, a3, a4, s], [a2, a3, a4, s], [a0, a2, a3, a4, s], [a1, a2, a3, a4, s], [a0, a1, a2, a3, a4, s]], minimum_cut_is_selected_from_the_complete_cut_family)
Retain all cut rows, the least exact capacity and every minimizing source-side set. A single small cut without complete comparison does not establish a minimum. The finite network certificate does not prove the general theorem. The exact result, retaining the requested order, is ([[[s], [[[s, a0], 9], [[s, a1], 10], [[s, a2], 11], [[s, a3], 12], [[s, a4], 13]], 55], [[a0, s], [[[a0, t], 9], [[s, a1], 10], [[s, a2], 11], [[s, a3], 12], [[s, a4], 13]], 55], [[a1, s], [[[a1, t], 10], [[s, a0], 9], [[s, a2], 11], [[s, a3], 12], [[s, a4], 13]], 55], [[a0, a1, s], [[[a0, t], 9], [[a1, t], 10], [[s, a2], 11], [[s, a3], 12], [[s, a4], 13]], 55], [[a2, s], [[[a2, t], 11], [[s, a0], 9], [[s, a1], 10], [[s, a3], 12], [[s, a4], 13]], 55], [[a0, a2, s], [[[a0, t], 9], [[a2, t], 11], [[s, a1], 10], [[s, a3], 12], [[s, a4], 13]], 55], [[a1, a2, s], [[[a1, t], 10], [[a2, t], 11], [[s, a0], 9], [[s, a3], 12], [[s, a4], 13]], 55], [[a0, a1, a2, s], [[[a0, t], 9], [[a1, t], 10], [[a2, t], 11], [[s, a3], 12], [[s, a4], 13]], 55], [[a3, s], [[[a3, t], 12], [[s, a0], 9], [[s, a1], 10], [[s, a2], 11], [[s, a4], 13]], 55], [[a0, a3, s], [[[a0, t], 9], [[a3, t], 12], [[s, a1], 10], [[s, a2], 11], [[s, a4], 13]], 55], [[a1, a3, s], [[[a1, t], 10], [[a3, t], 12], [[s, a0], 9], [[s, a2], 11], [[s, a4], 13]], 55], [[a0, a1, a3, s], [[[a0, t], 9], [[a1, t], 10], [[a3, t], 12], [[s, a2], 11], [[s, a4], 13]], 55], [[a2, a3, s], [[[a2, t], 11], [[a3, t], 12], [[s, a0], 9], [[s, a1], 10], [[s, a4], 13]], 55], [[a0, a2, a3, s], [[[a0, t], 9], [[a2, t], 11], [[a3, t], 12], [[s, a1], 10], [[s, a4], 13]], 55], [[a1, a2, a3, s], [[[a1, t], 10], [[a2, t], 11], [[a3, t], 12], [[s, a0], 9], [[s, a4], 13]], 55], [[a0, a1, a2, a3, s], [[[a0, t], 9], [[a1, t], 10], [[a2, t], 11], [[a3, t], 12], [[s, a4], 13]], 55], [[a4, s], [[[a4, t], 13], [[s, a0], 9], [[s, a1], 10], [[s, a2], 11], [[s, a3], 12]], 55], [[a0, a4, s], [[[a0, t], 9], [[a4, t], 13], [[s, a1], 10], [[s, a2], 11], [[s, a3], 12]], 55], [[a1, a4, s], [[[a1, t], 10], [[a4, t], 13], [[s, a0], 9], [[s, a2], 11], [[s, a3], 12]], 55], [[a0, a1, a4, s], [[[a0, t], 9], [[a1, t], 10], [[a4, t], 13], [[s, a2], 11], [[s, a3], 12]], 55], [[a2, a4, s], [[[a2, t], 11], [[a4, t], 13], [[s, a0], 9], [[s, a1], 10], [[s, a3], 12]], 55], [[a0, a2, a4, s], [[[a0, t], 9], [[a2, t], 11], [[a4, t], 13], [[s, a1], 10], [[s, a3], 12]], 55], [[a1, a2, a4, s], [[[a1, t], 10], [[a2, t], 11], [[a4, t], 13], [[s, a0], 9], [[s, a3], 12]], 55], [[a0, a1, a2, a4, s], [[[a0, t], 9], [[a1, t], 10], [[a2, t], 11], [[a4, t], 13], [[s, a3], 12]], 55], [[a3, a4, s], [[[a3, t], 12], [[a4, t], 13], [[s, a0], 9], [[s, a1], 10], [[s, a2], 11]], 55], [[a0, a3, a4, s], [[[a0, t], 9], [[a3, t], 12], [[a4, t], 13], [[s, a1], 10], [[s, a2], 11]], 55], [[a1, a3, a4, s], [[[a1, t], 10], [[a3, t], 12], [[a4, t], 13], [[s, a0], 9], [[s, a2], 11]], 55], [[a0, a1, a3, a4, s], [[[a0, t], 9], [[a1, t], 10], [[a3, t], 12], [[a4, t], 13], [[s, a2], 11]], 55], [[a2, a3, a4, s], [[[a2, t], 11], [[a3, t], 12], [[a4, t], 13], [[s, a0], 9], [[s, a1], 10]], 55], [[a0, a2, a3, a4, s], [[[a0, t], 9], [[a2, t], 11], [[a3, t], 12], [[a4, t], 13], [[s, a1], 10]], 55], [[a1, a2, a3, a4, s], [[[a1, t], 10], [[a2, t], 11], [[a3, t], 12], [[a4, t], 13], [[s, a0], 9]], 55], [[a0, a1, a2, a3, a4, s], [[[a0, t], 9], [[a1, t], 10], [[a2, t], 11], [[a3, t], 12], [[a4, t], 13]], 55]], 55, [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s], [a3, s], [a0, a3, s], [a1, a3, s], [a0, a1, a3, s], [a2, a3, s], [a0, a2, a3, s], [a1, a2, a3, s], [a0, a1, a2, a3, s], [a4, s], [a0, a4, s], [a1, a4, s], [a0, a1, a4, s], [a2, a4, s], [a0, a2, a4, s], [a1, a2, a4, s], [a0, a1, a2, a4, s], [a3, a4, s], [a0, a3, a4, s], [a1, a3, a4, s], [a0, a1, a3, a4, s], [a2, a3, a4, s], [a0, a2, a3, a4, s], [a1, a2, a3, a4, s], [a0, a1, a2, a3, a4, s]], minimum_cut_is_selected_from_the_complete_cut_family).
Reviewed step-by-step derivation
Max-flow min-cut theorem · qf-mod46-mfmc-minimum-cut-audit · candidate
59. The Minimum Cut Is Selected from the Complete Cut Family
Use complete directed vertices=[s, a0, a1, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t]], source=s, sink=t, aligned nonnegative exact capacities=[10, 10, 11, 11]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (cut_rows, minimum_capacity, minimum_cuts, certificate).
View answer and explanation
([[[s], [[[s, a0], 10], [[s, a1], 11]], 21], [[a0, s], [[[a0, t], 10], [[s, a1], 11]], 21], [[a1, s], [[[a1, t], 11], [[s, a0], 10]], 21], [[a0, a1, s], [[[a0, t], 10], [[a1, t], 11]], 21]], 21, [[s], [a0, s], [a1, s], [a0, a1, s]], minimum_cut_is_selected_from_the_complete_cut_family)
Retain all cut rows, the least exact capacity and every minimizing source-side set. A single small cut without complete comparison does not establish a minimum. The finite network certificate does not prove the general theorem. The exact result, retaining the requested order, is ([[[s], [[[s, a0], 10], [[s, a1], 11]], 21], [[a0, s], [[[a0, t], 10], [[s, a1], 11]], 21], [[a1, s], [[[a1, t], 11], [[s, a0], 10]], 21], [[a0, a1, s], [[[a0, t], 10], [[a1, t], 11]], 21]], 21, [[s], [a0, s], [a1, s], [a0, a1, s]], minimum_cut_is_selected_from_the_complete_cut_family).
Reviewed step-by-step derivation
Max-flow min-cut theorem · qf-mod46-mfmc-minimum-cut-audit · candidate
60. The Minimum Cut Is Selected from the Complete Cut Family
Use complete directed vertices=[s, a0, a1, a2, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t], [s, a2], [a2, t], [a0, s]], source=s, sink=t, aligned nonnegative exact capacities=[11, 11, 12, 12, 13, 13, 16]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (cut_rows, minimum_capacity, minimum_cuts, certificate).
View answer and explanation
([[[s], [[[s, a0], 11], [[s, a1], 12], [[s, a2], 13]], 36], [[a0, s], [[[a0, t], 11], [[s, a1], 12], [[s, a2], 13]], 36], [[a1, s], [[[a1, t], 12], [[s, a0], 11], [[s, a2], 13]], 36], [[a0, a1, s], [[[a0, t], 11], [[a1, t], 12], [[s, a2], 13]], 36], [[a2, s], [[[a2, t], 13], [[s, a0], 11], [[s, a1], 12]], 36], [[a0, a2, s], [[[a0, t], 11], [[a2, t], 13], [[s, a1], 12]], 36], [[a1, a2, s], [[[a1, t], 12], [[a2, t], 13], [[s, a0], 11]], 36], [[a0, a1, a2, s], [[[a0, t], 11], [[a1, t], 12], [[a2, t], 13]], 36]], 36, [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s]], minimum_cut_is_selected_from_the_complete_cut_family)
Retain all cut rows, the least exact capacity and every minimizing source-side set. A single small cut without complete comparison does not establish a minimum. The finite network certificate does not prove the general theorem. The exact result, retaining the requested order, is ([[[s], [[[s, a0], 11], [[s, a1], 12], [[s, a2], 13]], 36], [[a0, s], [[[a0, t], 11], [[s, a1], 12], [[s, a2], 13]], 36], [[a1, s], [[[a1, t], 12], [[s, a0], 11], [[s, a2], 13]], 36], [[a0, a1, s], [[[a0, t], 11], [[a1, t], 12], [[s, a2], 13]], 36], [[a2, s], [[[a2, t], 13], [[s, a0], 11], [[s, a1], 12]], 36], [[a0, a2, s], [[[a0, t], 11], [[a2, t], 13], [[s, a1], 12]], 36], [[a1, a2, s], [[[a1, t], 12], [[a2, t], 13], [[s, a0], 11]], 36], [[a0, a1, a2, s], [[[a0, t], 11], [[a1, t], 12], [[a2, t], 13]], 36]], 36, [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s]], minimum_cut_is_selected_from_the_complete_cut_family).
Reviewed step-by-step derivation
Max-flow min-cut theorem · qf-mod46-mfmc-minimum-cut-audit · candidate
61. The Minimum Cut Is Selected from the Complete Cut Family
Use complete directed vertices=[s, a0, a1, a2, a3, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t], [s, a2], [a2, t], [s, a3], [a3, t]], source=s, sink=t, aligned nonnegative exact capacities=[12, 12, 13, 13, 14, 14, 15, 15]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (cut_rows, minimum_capacity, minimum_cuts, certificate).
View answer and explanation
([[[s], [[[s, a0], 12], [[s, a1], 13], [[s, a2], 14], [[s, a3], 15]], 54], [[a0, s], [[[a0, t], 12], [[s, a1], 13], [[s, a2], 14], [[s, a3], 15]], 54], [[a1, s], [[[a1, t], 13], [[s, a0], 12], [[s, a2], 14], [[s, a3], 15]], 54], [[a0, a1, s], [[[a0, t], 12], [[a1, t], 13], [[s, a2], 14], [[s, a3], 15]], 54], [[a2, s], [[[a2, t], 14], [[s, a0], 12], [[s, a1], 13], [[s, a3], 15]], 54], [[a0, a2, s], [[[a0, t], 12], [[a2, t], 14], [[s, a1], 13], [[s, a3], 15]], 54], [[a1, a2, s], [[[a1, t], 13], [[a2, t], 14], [[s, a0], 12], [[s, a3], 15]], 54], [[a0, a1, a2, s], [[[a0, t], 12], [[a1, t], 13], [[a2, t], 14], [[s, a3], 15]], 54], [[a3, s], [[[a3, t], 15], [[s, a0], 12], [[s, a1], 13], [[s, a2], 14]], 54], [[a0, a3, s], [[[a0, t], 12], [[a3, t], 15], [[s, a1], 13], [[s, a2], 14]], 54], [[a1, a3, s], [[[a1, t], 13], [[a3, t], 15], [[s, a0], 12], [[s, a2], 14]], 54], [[a0, a1, a3, s], [[[a0, t], 12], [[a1, t], 13], [[a3, t], 15], [[s, a2], 14]], 54], [[a2, a3, s], [[[a2, t], 14], [[a3, t], 15], [[s, a0], 12], [[s, a1], 13]], 54], [[a0, a2, a3, s], [[[a0, t], 12], [[a2, t], 14], [[a3, t], 15], [[s, a1], 13]], 54], [[a1, a2, a3, s], [[[a1, t], 13], [[a2, t], 14], [[a3, t], 15], [[s, a0], 12]], 54], [[a0, a1, a2, a3, s], [[[a0, t], 12], [[a1, t], 13], [[a2, t], 14], [[a3, t], 15]], 54]], 54, [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s], [a3, s], [a0, a3, s], [a1, a3, s], [a0, a1, a3, s], [a2, a3, s], [a0, a2, a3, s], [a1, a2, a3, s], [a0, a1, a2, a3, s]], minimum_cut_is_selected_from_the_complete_cut_family)
Retain all cut rows, the least exact capacity and every minimizing source-side set. A single small cut without complete comparison does not establish a minimum. The finite network certificate does not prove the general theorem. The exact result, retaining the requested order, is ([[[s], [[[s, a0], 12], [[s, a1], 13], [[s, a2], 14], [[s, a3], 15]], 54], [[a0, s], [[[a0, t], 12], [[s, a1], 13], [[s, a2], 14], [[s, a3], 15]], 54], [[a1, s], [[[a1, t], 13], [[s, a0], 12], [[s, a2], 14], [[s, a3], 15]], 54], [[a0, a1, s], [[[a0, t], 12], [[a1, t], 13], [[s, a2], 14], [[s, a3], 15]], 54], [[a2, s], [[[a2, t], 14], [[s, a0], 12], [[s, a1], 13], [[s, a3], 15]], 54], [[a0, a2, s], [[[a0, t], 12], [[a2, t], 14], [[s, a1], 13], [[s, a3], 15]], 54], [[a1, a2, s], [[[a1, t], 13], [[a2, t], 14], [[s, a0], 12], [[s, a3], 15]], 54], [[a0, a1, a2, s], [[[a0, t], 12], [[a1, t], 13], [[a2, t], 14], [[s, a3], 15]], 54], [[a3, s], [[[a3, t], 15], [[s, a0], 12], [[s, a1], 13], [[s, a2], 14]], 54], [[a0, a3, s], [[[a0, t], 12], [[a3, t], 15], [[s, a1], 13], [[s, a2], 14]], 54], [[a1, a3, s], [[[a1, t], 13], [[a3, t], 15], [[s, a0], 12], [[s, a2], 14]], 54], [[a0, a1, a3, s], [[[a0, t], 12], [[a1, t], 13], [[a3, t], 15], [[s, a2], 14]], 54], [[a2, a3, s], [[[a2, t], 14], [[a3, t], 15], [[s, a0], 12], [[s, a1], 13]], 54], [[a0, a2, a3, s], [[[a0, t], 12], [[a2, t], 14], [[a3, t], 15], [[s, a1], 13]], 54], [[a1, a2, a3, s], [[[a1, t], 13], [[a2, t], 14], [[a3, t], 15], [[s, a0], 12]], 54], [[a0, a1, a2, a3, s], [[[a0, t], 12], [[a1, t], 13], [[a2, t], 14], [[a3, t], 15]], 54]], 54, [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s], [a3, s], [a0, a3, s], [a1, a3, s], [a0, a1, a3, s], [a2, a3, s], [a0, a2, a3, s], [a1, a2, a3, s], [a0, a1, a2, a3, s]], minimum_cut_is_selected_from_the_complete_cut_family).
Reviewed step-by-step derivation
Max-flow min-cut theorem · qf-mod46-mfmc-minimum-cut-audit · candidate
62. The Minimum Cut Is Selected from the Complete Cut Family
Use complete directed vertices=[s, a0, a1, a2, a3, a4, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t], [s, a2], [a2, t], [s, a3], [a3, t], [s, a4], [a4, t]], source=s, sink=t, aligned nonnegative exact capacities=[13, 13, 14, 14, 15, 15, 16, 16, 17, 17]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (cut_rows, minimum_capacity, minimum_cuts, certificate).
View answer and explanation
([[[s], [[[s, a0], 13], [[s, a1], 14], [[s, a2], 15], [[s, a3], 16], [[s, a4], 17]], 75], [[a0, s], [[[a0, t], 13], [[s, a1], 14], [[s, a2], 15], [[s, a3], 16], [[s, a4], 17]], 75], [[a1, s], [[[a1, t], 14], [[s, a0], 13], [[s, a2], 15], [[s, a3], 16], [[s, a4], 17]], 75], [[a0, a1, s], [[[a0, t], 13], [[a1, t], 14], [[s, a2], 15], [[s, a3], 16], [[s, a4], 17]], 75], [[a2, s], [[[a2, t], 15], [[s, a0], 13], [[s, a1], 14], [[s, a3], 16], [[s, a4], 17]], 75], [[a0, a2, s], [[[a0, t], 13], [[a2, t], 15], [[s, a1], 14], [[s, a3], 16], [[s, a4], 17]], 75], [[a1, a2, s], [[[a1, t], 14], [[a2, t], 15], [[s, a0], 13], [[s, a3], 16], [[s, a4], 17]], 75], [[a0, a1, a2, s], [[[a0, t], 13], [[a1, t], 14], [[a2, t], 15], [[s, a3], 16], [[s, a4], 17]], 75], [[a3, s], [[[a3, t], 16], [[s, a0], 13], [[s, a1], 14], [[s, a2], 15], [[s, a4], 17]], 75], [[a0, a3, s], [[[a0, t], 13], [[a3, t], 16], [[s, a1], 14], [[s, a2], 15], [[s, a4], 17]], 75], [[a1, a3, s], [[[a1, t], 14], [[a3, t], 16], [[s, a0], 13], [[s, a2], 15], [[s, a4], 17]], 75], [[a0, a1, a3, s], [[[a0, t], 13], [[a1, t], 14], [[a3, t], 16], [[s, a2], 15], [[s, a4], 17]], 75], [[a2, a3, s], [[[a2, t], 15], [[a3, t], 16], [[s, a0], 13], [[s, a1], 14], [[s, a4], 17]], 75], [[a0, a2, a3, s], [[[a0, t], 13], [[a2, t], 15], [[a3, t], 16], [[s, a1], 14], [[s, a4], 17]], 75], [[a1, a2, a3, s], [[[a1, t], 14], [[a2, t], 15], [[a3, t], 16], [[s, a0], 13], [[s, a4], 17]], 75], [[a0, a1, a2, a3, s], [[[a0, t], 13], [[a1, t], 14], [[a2, t], 15], [[a3, t], 16], [[s, a4], 17]], 75], [[a4, s], [[[a4, t], 17], [[s, a0], 13], [[s, a1], 14], [[s, a2], 15], [[s, a3], 16]], 75], [[a0, a4, s], [[[a0, t], 13], [[a4, t], 17], [[s, a1], 14], [[s, a2], 15], [[s, a3], 16]], 75], [[a1, a4, s], [[[a1, t], 14], [[a4, t], 17], [[s, a0], 13], [[s, a2], 15], [[s, a3], 16]], 75], [[a0, a1, a4, s], [[[a0, t], 13], [[a1, t], 14], [[a4, t], 17], [[s, a2], 15], [[s, a3], 16]], 75], [[a2, a4, s], [[[a2, t], 15], [[a4, t], 17], [[s, a0], 13], [[s, a1], 14], [[s, a3], 16]], 75], [[a0, a2, a4, s], [[[a0, t], 13], [[a2, t], 15], [[a4, t], 17], [[s, a1], 14], [[s, a3], 16]], 75], [[a1, a2, a4, s], [[[a1, t], 14], [[a2, t], 15], [[a4, t], 17], [[s, a0], 13], [[s, a3], 16]], 75], [[a0, a1, a2, a4, s], [[[a0, t], 13], [[a1, t], 14], [[a2, t], 15], [[a4, t], 17], [[s, a3], 16]], 75], [[a3, a4, s], [[[a3, t], 16], [[a4, t], 17], [[s, a0], 13], [[s, a1], 14], [[s, a2], 15]], 75], [[a0, a3, a4, s], [[[a0, t], 13], [[a3, t], 16], [[a4, t], 17], [[s, a1], 14], [[s, a2], 15]], 75], [[a1, a3, a4, s], [[[a1, t], 14], [[a3, t], 16], [[a4, t], 17], [[s, a0], 13], [[s, a2], 15]], 75], [[a0, a1, a3, a4, s], [[[a0, t], 13], [[a1, t], 14], [[a3, t], 16], [[a4, t], 17], [[s, a2], 15]], 75], [[a2, a3, a4, s], [[[a2, t], 15], [[a3, t], 16], [[a4, t], 17], [[s, a0], 13], [[s, a1], 14]], 75], [[a0, a2, a3, a4, s], [[[a0, t], 13], [[a2, t], 15], [[a3, t], 16], [[a4, t], 17], [[s, a1], 14]], 75], [[a1, a2, a3, a4, s], [[[a1, t], 14], [[a2, t], 15], [[a3, t], 16], [[a4, t], 17], [[s, a0], 13]], 75], [[a0, a1, a2, a3, a4, s], [[[a0, t], 13], [[a1, t], 14], [[a2, t], 15], [[a3, t], 16], [[a4, t], 17]], 75]], 75, [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s], [a3, s], [a0, a3, s], [a1, a3, s], [a0, a1, a3, s], [a2, a3, s], [a0, a2, a3, s], [a1, a2, a3, s], [a0, a1, a2, a3, s], [a4, s], [a0, a4, s], [a1, a4, s], [a0, a1, a4, s], [a2, a4, s], [a0, a2, a4, s], [a1, a2, a4, s], [a0, a1, a2, a4, s], [a3, a4, s], [a0, a3, a4, s], [a1, a3, a4, s], [a0, a1, a3, a4, s], [a2, a3, a4, s], [a0, a2, a3, a4, s], [a1, a2, a3, a4, s], [a0, a1, a2, a3, a4, s]], minimum_cut_is_selected_from_the_complete_cut_family)
Retain all cut rows, the least exact capacity and every minimizing source-side set. A single small cut without complete comparison does not establish a minimum. The finite network certificate does not prove the general theorem. The exact result, retaining the requested order, is ([[[s], [[[s, a0], 13], [[s, a1], 14], [[s, a2], 15], [[s, a3], 16], [[s, a4], 17]], 75], [[a0, s], [[[a0, t], 13], [[s, a1], 14], [[s, a2], 15], [[s, a3], 16], [[s, a4], 17]], 75], [[a1, s], [[[a1, t], 14], [[s, a0], 13], [[s, a2], 15], [[s, a3], 16], [[s, a4], 17]], 75], [[a0, a1, s], [[[a0, t], 13], [[a1, t], 14], [[s, a2], 15], [[s, a3], 16], [[s, a4], 17]], 75], [[a2, s], [[[a2, t], 15], [[s, a0], 13], [[s, a1], 14], [[s, a3], 16], [[s, a4], 17]], 75], [[a0, a2, s], [[[a0, t], 13], [[a2, t], 15], [[s, a1], 14], [[s, a3], 16], [[s, a4], 17]], 75], [[a1, a2, s], [[[a1, t], 14], [[a2, t], 15], [[s, a0], 13], [[s, a3], 16], [[s, a4], 17]], 75], [[a0, a1, a2, s], [[[a0, t], 13], [[a1, t], 14], [[a2, t], 15], [[s, a3], 16], [[s, a4], 17]], 75], [[a3, s], [[[a3, t], 16], [[s, a0], 13], [[s, a1], 14], [[s, a2], 15], [[s, a4], 17]], 75], [[a0, a3, s], [[[a0, t], 13], [[a3, t], 16], [[s, a1], 14], [[s, a2], 15], [[s, a4], 17]], 75], [[a1, a3, s], [[[a1, t], 14], [[a3, t], 16], [[s, a0], 13], [[s, a2], 15], [[s, a4], 17]], 75], [[a0, a1, a3, s], [[[a0, t], 13], [[a1, t], 14], [[a3, t], 16], [[s, a2], 15], [[s, a4], 17]], 75], [[a2, a3, s], [[[a2, t], 15], [[a3, t], 16], [[s, a0], 13], [[s, a1], 14], [[s, a4], 17]], 75], [[a0, a2, a3, s], [[[a0, t], 13], [[a2, t], 15], [[a3, t], 16], [[s, a1], 14], [[s, a4], 17]], 75], [[a1, a2, a3, s], [[[a1, t], 14], [[a2, t], 15], [[a3, t], 16], [[s, a0], 13], [[s, a4], 17]], 75], [[a0, a1, a2, a3, s], [[[a0, t], 13], [[a1, t], 14], [[a2, t], 15], [[a3, t], 16], [[s, a4], 17]], 75], [[a4, s], [[[a4, t], 17], [[s, a0], 13], [[s, a1], 14], [[s, a2], 15], [[s, a3], 16]], 75], [[a0, a4, s], [[[a0, t], 13], [[a4, t], 17], [[s, a1], 14], [[s, a2], 15], [[s, a3], 16]], 75], [[a1, a4, s], [[[a1, t], 14], [[a4, t], 17], [[s, a0], 13], [[s, a2], 15], [[s, a3], 16]], 75], [[a0, a1, a4, s], [[[a0, t], 13], [[a1, t], 14], [[a4, t], 17], [[s, a2], 15], [[s, a3], 16]], 75], [[a2, a4, s], [[[a2, t], 15], [[a4, t], 17], [[s, a0], 13], [[s, a1], 14], [[s, a3], 16]], 75], [[a0, a2, a4, s], [[[a0, t], 13], [[a2, t], 15], [[a4, t], 17], [[s, a1], 14], [[s, a3], 16]], 75], [[a1, a2, a4, s], [[[a1, t], 14], [[a2, t], 15], [[a4, t], 17], [[s, a0], 13], [[s, a3], 16]], 75], [[a0, a1, a2, a4, s], [[[a0, t], 13], [[a1, t], 14], [[a2, t], 15], [[a4, t], 17], [[s, a3], 16]], 75], [[a3, a4, s], [[[a3, t], 16], [[a4, t], 17], [[s, a0], 13], [[s, a1], 14], [[s, a2], 15]], 75], [[a0, a3, a4, s], [[[a0, t], 13], [[a3, t], 16], [[a4, t], 17], [[s, a1], 14], [[s, a2], 15]], 75], [[a1, a3, a4, s], [[[a1, t], 14], [[a3, t], 16], [[a4, t], 17], [[s, a0], 13], [[s, a2], 15]], 75], [[a0, a1, a3, a4, s], [[[a0, t], 13], [[a1, t], 14], [[a3, t], 16], [[a4, t], 17], [[s, a2], 15]], 75], [[a2, a3, a4, s], [[[a2, t], 15], [[a3, t], 16], [[a4, t], 17], [[s, a0], 13], [[s, a1], 14]], 75], [[a0, a2, a3, a4, s], [[[a0, t], 13], [[a2, t], 15], [[a3, t], 16], [[a4, t], 17], [[s, a1], 14]], 75], [[a1, a2, a3, a4, s], [[[a1, t], 14], [[a2, t], 15], [[a3, t], 16], [[a4, t], 17], [[s, a0], 13]], 75], [[a0, a1, a2, a3, a4, s], [[[a0, t], 13], [[a1, t], 14], [[a2, t], 15], [[a3, t], 16], [[a4, t], 17]], 75]], 75, [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s], [a3, s], [a0, a3, s], [a1, a3, s], [a0, a1, a3, s], [a2, a3, s], [a0, a2, a3, s], [a1, a2, a3, s], [a0, a1, a2, a3, s], [a4, s], [a0, a4, s], [a1, a4, s], [a0, a1, a4, s], [a2, a4, s], [a0, a2, a4, s], [a1, a2, a4, s], [a0, a1, a2, a4, s], [a3, a4, s], [a0, a3, a4, s], [a1, a3, a4, s], [a0, a1, a3, a4, s], [a2, a3, a4, s], [a0, a2, a3, a4, s], [a1, a2, a3, a4, s], [a0, a1, a2, a3, a4, s]], minimum_cut_is_selected_from_the_complete_cut_family).
Reviewed step-by-step derivation
Max-flow min-cut theorem · qf-mod46-mfmc-minimum-cut-audit · candidate
63. The Minimum Cut Is Selected from the Complete Cut Family
Use complete directed vertices=[s, a0, a1, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t], [a0, s]], source=s, sink=t, aligned nonnegative exact capacities=[14, 14, 15, 15, 19]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (cut_rows, minimum_capacity, minimum_cuts, certificate).
View answer and explanation
([[[s], [[[s, a0], 14], [[s, a1], 15]], 29], [[a0, s], [[[a0, t], 14], [[s, a1], 15]], 29], [[a1, s], [[[a1, t], 15], [[s, a0], 14]], 29], [[a0, a1, s], [[[a0, t], 14], [[a1, t], 15]], 29]], 29, [[s], [a0, s], [a1, s], [a0, a1, s]], minimum_cut_is_selected_from_the_complete_cut_family)
Retain all cut rows, the least exact capacity and every minimizing source-side set. A single small cut without complete comparison does not establish a minimum. The finite network certificate does not prove the general theorem. The exact result, retaining the requested order, is ([[[s], [[[s, a0], 14], [[s, a1], 15]], 29], [[a0, s], [[[a0, t], 14], [[s, a1], 15]], 29], [[a1, s], [[[a1, t], 15], [[s, a0], 14]], 29], [[a0, a1, s], [[[a0, t], 14], [[a1, t], 15]], 29]], 29, [[s], [a0, s], [a1, s], [a0, a1, s]], minimum_cut_is_selected_from_the_complete_cut_family).
Reviewed step-by-step derivation
Max-flow min-cut theorem · qf-mod46-mfmc-minimum-cut-audit · candidate
64. The Minimum Cut Is Selected from the Complete Cut Family
Use complete directed vertices=[s, a0, a1, a2, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t], [s, a2], [a2, t]], source=s, sink=t, aligned nonnegative exact capacities=[15, 15, 16, 16, 17, 17]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (cut_rows, minimum_capacity, minimum_cuts, certificate).
View answer and explanation
([[[s], [[[s, a0], 15], [[s, a1], 16], [[s, a2], 17]], 48], [[a0, s], [[[a0, t], 15], [[s, a1], 16], [[s, a2], 17]], 48], [[a1, s], [[[a1, t], 16], [[s, a0], 15], [[s, a2], 17]], 48], [[a0, a1, s], [[[a0, t], 15], [[a1, t], 16], [[s, a2], 17]], 48], [[a2, s], [[[a2, t], 17], [[s, a0], 15], [[s, a1], 16]], 48], [[a0, a2, s], [[[a0, t], 15], [[a2, t], 17], [[s, a1], 16]], 48], [[a1, a2, s], [[[a1, t], 16], [[a2, t], 17], [[s, a0], 15]], 48], [[a0, a1, a2, s], [[[a0, t], 15], [[a1, t], 16], [[a2, t], 17]], 48]], 48, [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s]], minimum_cut_is_selected_from_the_complete_cut_family)
Retain all cut rows, the least exact capacity and every minimizing source-side set. A single small cut without complete comparison does not establish a minimum. The finite network certificate does not prove the general theorem. The exact result, retaining the requested order, is ([[[s], [[[s, a0], 15], [[s, a1], 16], [[s, a2], 17]], 48], [[a0, s], [[[a0, t], 15], [[s, a1], 16], [[s, a2], 17]], 48], [[a1, s], [[[a1, t], 16], [[s, a0], 15], [[s, a2], 17]], 48], [[a0, a1, s], [[[a0, t], 15], [[a1, t], 16], [[s, a2], 17]], 48], [[a2, s], [[[a2, t], 17], [[s, a0], 15], [[s, a1], 16]], 48], [[a0, a2, s], [[[a0, t], 15], [[a2, t], 17], [[s, a1], 16]], 48], [[a1, a2, s], [[[a1, t], 16], [[a2, t], 17], [[s, a0], 15]], 48], [[a0, a1, a2, s], [[[a0, t], 15], [[a1, t], 16], [[a2, t], 17]], 48]], 48, [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s]], minimum_cut_is_selected_from_the_complete_cut_family).
Reviewed step-by-step derivation
Max-flow min-cut theorem · qf-mod46-mfmc-minimum-cut-audit · candidate
65. The Minimum Cut Is Selected from the Complete Cut Family
Use complete directed vertices=[s, a0, a1, a2, a3, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t], [s, a2], [a2, t], [s, a3], [a3, t]], source=s, sink=t, aligned nonnegative exact capacities=[16, 16, 17, 17, 18, 18, 19, 19]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (cut_rows, minimum_capacity, minimum_cuts, certificate).
View answer and explanation
([[[s], [[[s, a0], 16], [[s, a1], 17], [[s, a2], 18], [[s, a3], 19]], 70], [[a0, s], [[[a0, t], 16], [[s, a1], 17], [[s, a2], 18], [[s, a3], 19]], 70], [[a1, s], [[[a1, t], 17], [[s, a0], 16], [[s, a2], 18], [[s, a3], 19]], 70], [[a0, a1, s], [[[a0, t], 16], [[a1, t], 17], [[s, a2], 18], [[s, a3], 19]], 70], [[a2, s], [[[a2, t], 18], [[s, a0], 16], [[s, a1], 17], [[s, a3], 19]], 70], [[a0, a2, s], [[[a0, t], 16], [[a2, t], 18], [[s, a1], 17], [[s, a3], 19]], 70], [[a1, a2, s], [[[a1, t], 17], [[a2, t], 18], [[s, a0], 16], [[s, a3], 19]], 70], [[a0, a1, a2, s], [[[a0, t], 16], [[a1, t], 17], [[a2, t], 18], [[s, a3], 19]], 70], [[a3, s], [[[a3, t], 19], [[s, a0], 16], [[s, a1], 17], [[s, a2], 18]], 70], [[a0, a3, s], [[[a0, t], 16], [[a3, t], 19], [[s, a1], 17], [[s, a2], 18]], 70], [[a1, a3, s], [[[a1, t], 17], [[a3, t], 19], [[s, a0], 16], [[s, a2], 18]], 70], [[a0, a1, a3, s], [[[a0, t], 16], [[a1, t], 17], [[a3, t], 19], [[s, a2], 18]], 70], [[a2, a3, s], [[[a2, t], 18], [[a3, t], 19], [[s, a0], 16], [[s, a1], 17]], 70], [[a0, a2, a3, s], [[[a0, t], 16], [[a2, t], 18], [[a3, t], 19], [[s, a1], 17]], 70], [[a1, a2, a3, s], [[[a1, t], 17], [[a2, t], 18], [[a3, t], 19], [[s, a0], 16]], 70], [[a0, a1, a2, a3, s], [[[a0, t], 16], [[a1, t], 17], [[a2, t], 18], [[a3, t], 19]], 70]], 70, [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s], [a3, s], [a0, a3, s], [a1, a3, s], [a0, a1, a3, s], [a2, a3, s], [a0, a2, a3, s], [a1, a2, a3, s], [a0, a1, a2, a3, s]], minimum_cut_is_selected_from_the_complete_cut_family)
Retain all cut rows, the least exact capacity and every minimizing source-side set. A single small cut without complete comparison does not establish a minimum. The finite network certificate does not prove the general theorem. The exact result, retaining the requested order, is ([[[s], [[[s, a0], 16], [[s, a1], 17], [[s, a2], 18], [[s, a3], 19]], 70], [[a0, s], [[[a0, t], 16], [[s, a1], 17], [[s, a2], 18], [[s, a3], 19]], 70], [[a1, s], [[[a1, t], 17], [[s, a0], 16], [[s, a2], 18], [[s, a3], 19]], 70], [[a0, a1, s], [[[a0, t], 16], [[a1, t], 17], [[s, a2], 18], [[s, a3], 19]], 70], [[a2, s], [[[a2, t], 18], [[s, a0], 16], [[s, a1], 17], [[s, a3], 19]], 70], [[a0, a2, s], [[[a0, t], 16], [[a2, t], 18], [[s, a1], 17], [[s, a3], 19]], 70], [[a1, a2, s], [[[a1, t], 17], [[a2, t], 18], [[s, a0], 16], [[s, a3], 19]], 70], [[a0, a1, a2, s], [[[a0, t], 16], [[a1, t], 17], [[a2, t], 18], [[s, a3], 19]], 70], [[a3, s], [[[a3, t], 19], [[s, a0], 16], [[s, a1], 17], [[s, a2], 18]], 70], [[a0, a3, s], [[[a0, t], 16], [[a3, t], 19], [[s, a1], 17], [[s, a2], 18]], 70], [[a1, a3, s], [[[a1, t], 17], [[a3, t], 19], [[s, a0], 16], [[s, a2], 18]], 70], [[a0, a1, a3, s], [[[a0, t], 16], [[a1, t], 17], [[a3, t], 19], [[s, a2], 18]], 70], [[a2, a3, s], [[[a2, t], 18], [[a3, t], 19], [[s, a0], 16], [[s, a1], 17]], 70], [[a0, a2, a3, s], [[[a0, t], 16], [[a2, t], 18], [[a3, t], 19], [[s, a1], 17]], 70], [[a1, a2, a3, s], [[[a1, t], 17], [[a2, t], 18], [[a3, t], 19], [[s, a0], 16]], 70], [[a0, a1, a2, a3, s], [[[a0, t], 16], [[a1, t], 17], [[a2, t], 18], [[a3, t], 19]], 70]], 70, [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s], [a3, s], [a0, a3, s], [a1, a3, s], [a0, a1, a3, s], [a2, a3, s], [a0, a2, a3, s], [a1, a2, a3, s], [a0, a1, a2, a3, s]], minimum_cut_is_selected_from_the_complete_cut_family).
Reviewed step-by-step derivation
Max-flow min-cut theorem · qf-mod46-mfmc-minimum-cut-audit · candidate
66. The Minimum Cut Is Selected from the Complete Cut Family
Use complete directed vertices=[s, a0, a1, a2, a3, a4, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t], [s, a2], [a2, t], [s, a3], [a3, t], [s, a4], [a4, t], [a0, s]], source=s, sink=t, aligned nonnegative exact capacities=[17, 17, 18, 18, 19, 19, 20, 20, 21, 21, 22]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (cut_rows, minimum_capacity, minimum_cuts, certificate).
View answer and explanation
([[[s], [[[s, a0], 17], [[s, a1], 18], [[s, a2], 19], [[s, a3], 20], [[s, a4], 21]], 95], [[a0, s], [[[a0, t], 17], [[s, a1], 18], [[s, a2], 19], [[s, a3], 20], [[s, a4], 21]], 95], [[a1, s], [[[a1, t], 18], [[s, a0], 17], [[s, a2], 19], [[s, a3], 20], [[s, a4], 21]], 95], [[a0, a1, s], [[[a0, t], 17], [[a1, t], 18], [[s, a2], 19], [[s, a3], 20], [[s, a4], 21]], 95], [[a2, s], [[[a2, t], 19], [[s, a0], 17], [[s, a1], 18], [[s, a3], 20], [[s, a4], 21]], 95], [[a0, a2, s], [[[a0, t], 17], [[a2, t], 19], [[s, a1], 18], [[s, a3], 20], [[s, a4], 21]], 95], [[a1, a2, s], [[[a1, t], 18], [[a2, t], 19], [[s, a0], 17], [[s, a3], 20], [[s, a4], 21]], 95], [[a0, a1, a2, s], [[[a0, t], 17], [[a1, t], 18], [[a2, t], 19], [[s, a3], 20], [[s, a4], 21]], 95], [[a3, s], [[[a3, t], 20], [[s, a0], 17], [[s, a1], 18], [[s, a2], 19], [[s, a4], 21]], 95], [[a0, a3, s], [[[a0, t], 17], [[a3, t], 20], [[s, a1], 18], [[s, a2], 19], [[s, a4], 21]], 95], [[a1, a3, s], [[[a1, t], 18], [[a3, t], 20], [[s, a0], 17], [[s, a2], 19], [[s, a4], 21]], 95], [[a0, a1, a3, s], [[[a0, t], 17], [[a1, t], 18], [[a3, t], 20], [[s, a2], 19], [[s, a4], 21]], 95], [[a2, a3, s], [[[a2, t], 19], [[a3, t], 20], [[s, a0], 17], [[s, a1], 18], [[s, a4], 21]], 95], [[a0, a2, a3, s], [[[a0, t], 17], [[a2, t], 19], [[a3, t], 20], [[s, a1], 18], [[s, a4], 21]], 95], [[a1, a2, a3, s], [[[a1, t], 18], [[a2, t], 19], [[a3, t], 20], [[s, a0], 17], [[s, a4], 21]], 95], [[a0, a1, a2, a3, s], [[[a0, t], 17], [[a1, t], 18], [[a2, t], 19], [[a3, t], 20], [[s, a4], 21]], 95], [[a4, s], [[[a4, t], 21], [[s, a0], 17], [[s, a1], 18], [[s, a2], 19], [[s, a3], 20]], 95], [[a0, a4, s], [[[a0, t], 17], [[a4, t], 21], [[s, a1], 18], [[s, a2], 19], [[s, a3], 20]], 95], [[a1, a4, s], [[[a1, t], 18], [[a4, t], 21], [[s, a0], 17], [[s, a2], 19], [[s, a3], 20]], 95], [[a0, a1, a4, s], [[[a0, t], 17], [[a1, t], 18], [[a4, t], 21], [[s, a2], 19], [[s, a3], 20]], 95], [[a2, a4, s], [[[a2, t], 19], [[a4, t], 21], [[s, a0], 17], [[s, a1], 18], [[s, a3], 20]], 95], [[a0, a2, a4, s], [[[a0, t], 17], [[a2, t], 19], [[a4, t], 21], [[s, a1], 18], [[s, a3], 20]], 95], [[a1, a2, a4, s], [[[a1, t], 18], [[a2, t], 19], [[a4, t], 21], [[s, a0], 17], [[s, a3], 20]], 95], [[a0, a1, a2, a4, s], [[[a0, t], 17], [[a1, t], 18], [[a2, t], 19], [[a4, t], 21], [[s, a3], 20]], 95], [[a3, a4, s], [[[a3, t], 20], [[a4, t], 21], [[s, a0], 17], [[s, a1], 18], [[s, a2], 19]], 95], [[a0, a3, a4, s], [[[a0, t], 17], [[a3, t], 20], [[a4, t], 21], [[s, a1], 18], [[s, a2], 19]], 95], [[a1, a3, a4, s], [[[a1, t], 18], [[a3, t], 20], [[a4, t], 21], [[s, a0], 17], [[s, a2], 19]], 95], [[a0, a1, a3, a4, s], [[[a0, t], 17], [[a1, t], 18], [[a3, t], 20], [[a4, t], 21], [[s, a2], 19]], 95], [[a2, a3, a4, s], [[[a2, t], 19], [[a3, t], 20], [[a4, t], 21], [[s, a0], 17], [[s, a1], 18]], 95], [[a0, a2, a3, a4, s], [[[a0, t], 17], [[a2, t], 19], [[a3, t], 20], [[a4, t], 21], [[s, a1], 18]], 95], [[a1, a2, a3, a4, s], [[[a1, t], 18], [[a2, t], 19], [[a3, t], 20], [[a4, t], 21], [[s, a0], 17]], 95], [[a0, a1, a2, a3, a4, s], [[[a0, t], 17], [[a1, t], 18], [[a2, t], 19], [[a3, t], 20], [[a4, t], 21]], 95]], 95, [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s], [a3, s], [a0, a3, s], [a1, a3, s], [a0, a1, a3, s], [a2, a3, s], [a0, a2, a3, s], [a1, a2, a3, s], [a0, a1, a2, a3, s], [a4, s], [a0, a4, s], [a1, a4, s], [a0, a1, a4, s], [a2, a4, s], [a0, a2, a4, s], [a1, a2, a4, s], [a0, a1, a2, a4, s], [a3, a4, s], [a0, a3, a4, s], [a1, a3, a4, s], [a0, a1, a3, a4, s], [a2, a3, a4, s], [a0, a2, a3, a4, s], [a1, a2, a3, a4, s], [a0, a1, a2, a3, a4, s]], minimum_cut_is_selected_from_the_complete_cut_family)
Retain all cut rows, the least exact capacity and every minimizing source-side set. A single small cut without complete comparison does not establish a minimum. The finite network certificate does not prove the general theorem. The exact result, retaining the requested order, is ([[[s], [[[s, a0], 17], [[s, a1], 18], [[s, a2], 19], [[s, a3], 20], [[s, a4], 21]], 95], [[a0, s], [[[a0, t], 17], [[s, a1], 18], [[s, a2], 19], [[s, a3], 20], [[s, a4], 21]], 95], [[a1, s], [[[a1, t], 18], [[s, a0], 17], [[s, a2], 19], [[s, a3], 20], [[s, a4], 21]], 95], [[a0, a1, s], [[[a0, t], 17], [[a1, t], 18], [[s, a2], 19], [[s, a3], 20], [[s, a4], 21]], 95], [[a2, s], [[[a2, t], 19], [[s, a0], 17], [[s, a1], 18], [[s, a3], 20], [[s, a4], 21]], 95], [[a0, a2, s], [[[a0, t], 17], [[a2, t], 19], [[s, a1], 18], [[s, a3], 20], [[s, a4], 21]], 95], [[a1, a2, s], [[[a1, t], 18], [[a2, t], 19], [[s, a0], 17], [[s, a3], 20], [[s, a4], 21]], 95], [[a0, a1, a2, s], [[[a0, t], 17], [[a1, t], 18], [[a2, t], 19], [[s, a3], 20], [[s, a4], 21]], 95], [[a3, s], [[[a3, t], 20], [[s, a0], 17], [[s, a1], 18], [[s, a2], 19], [[s, a4], 21]], 95], [[a0, a3, s], [[[a0, t], 17], [[a3, t], 20], [[s, a1], 18], [[s, a2], 19], [[s, a4], 21]], 95], [[a1, a3, s], [[[a1, t], 18], [[a3, t], 20], [[s, a0], 17], [[s, a2], 19], [[s, a4], 21]], 95], [[a0, a1, a3, s], [[[a0, t], 17], [[a1, t], 18], [[a3, t], 20], [[s, a2], 19], [[s, a4], 21]], 95], [[a2, a3, s], [[[a2, t], 19], [[a3, t], 20], [[s, a0], 17], [[s, a1], 18], [[s, a4], 21]], 95], [[a0, a2, a3, s], [[[a0, t], 17], [[a2, t], 19], [[a3, t], 20], [[s, a1], 18], [[s, a4], 21]], 95], [[a1, a2, a3, s], [[[a1, t], 18], [[a2, t], 19], [[a3, t], 20], [[s, a0], 17], [[s, a4], 21]], 95], [[a0, a1, a2, a3, s], [[[a0, t], 17], [[a1, t], 18], [[a2, t], 19], [[a3, t], 20], [[s, a4], 21]], 95], [[a4, s], [[[a4, t], 21], [[s, a0], 17], [[s, a1], 18], [[s, a2], 19], [[s, a3], 20]], 95], [[a0, a4, s], [[[a0, t], 17], [[a4, t], 21], [[s, a1], 18], [[s, a2], 19], [[s, a3], 20]], 95], [[a1, a4, s], [[[a1, t], 18], [[a4, t], 21], [[s, a0], 17], [[s, a2], 19], [[s, a3], 20]], 95], [[a0, a1, a4, s], [[[a0, t], 17], [[a1, t], 18], [[a4, t], 21], [[s, a2], 19], [[s, a3], 20]], 95], [[a2, a4, s], [[[a2, t], 19], [[a4, t], 21], [[s, a0], 17], [[s, a1], 18], [[s, a3], 20]], 95], [[a0, a2, a4, s], [[[a0, t], 17], [[a2, t], 19], [[a4, t], 21], [[s, a1], 18], [[s, a3], 20]], 95], [[a1, a2, a4, s], [[[a1, t], 18], [[a2, t], 19], [[a4, t], 21], [[s, a0], 17], [[s, a3], 20]], 95], [[a0, a1, a2, a4, s], [[[a0, t], 17], [[a1, t], 18], [[a2, t], 19], [[a4, t], 21], [[s, a3], 20]], 95], [[a3, a4, s], [[[a3, t], 20], [[a4, t], 21], [[s, a0], 17], [[s, a1], 18], [[s, a2], 19]], 95], [[a0, a3, a4, s], [[[a0, t], 17], [[a3, t], 20], [[a4, t], 21], [[s, a1], 18], [[s, a2], 19]], 95], [[a1, a3, a4, s], [[[a1, t], 18], [[a3, t], 20], [[a4, t], 21], [[s, a0], 17], [[s, a2], 19]], 95], [[a0, a1, a3, a4, s], [[[a0, t], 17], [[a1, t], 18], [[a3, t], 20], [[a4, t], 21], [[s, a2], 19]], 95], [[a2, a3, a4, s], [[[a2, t], 19], [[a3, t], 20], [[a4, t], 21], [[s, a0], 17], [[s, a1], 18]], 95], [[a0, a2, a3, a4, s], [[[a0, t], 17], [[a2, t], 19], [[a3, t], 20], [[a4, t], 21], [[s, a1], 18]], 95], [[a1, a2, a3, a4, s], [[[a1, t], 18], [[a2, t], 19], [[a3, t], 20], [[a4, t], 21], [[s, a0], 17]], 95], [[a0, a1, a2, a3, a4, s], [[[a0, t], 17], [[a1, t], 18], [[a2, t], 19], [[a3, t], 20], [[a4, t], 21]], 95]], 95, [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s], [a3, s], [a0, a3, s], [a1, a3, s], [a0, a1, a3, s], [a2, a3, s], [a0, a2, a3, s], [a1, a2, a3, s], [a0, a1, a2, a3, s], [a4, s], [a0, a4, s], [a1, a4, s], [a0, a1, a4, s], [a2, a4, s], [a0, a2, a4, s], [a1, a2, a4, s], [a0, a1, a2, a4, s], [a3, a4, s], [a0, a3, a4, s], [a1, a3, a4, s], [a0, a1, a3, a4, s], [a2, a3, a4, s], [a0, a2, a3, a4, s], [a1, a2, a3, a4, s], [a0, a1, a2, a3, a4, s]], minimum_cut_is_selected_from_the_complete_cut_family).
Reviewed step-by-step derivation
Max-flow min-cut theorem · qf-mod46-mfmc-minimum-cut-audit · candidate
67. The Minimum Cut Is Selected from the Complete Cut Family
Use complete directed vertices=[s, a0, a1, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t]], source=s, sink=t, aligned nonnegative exact capacities=[18, 18, 19, 19]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (cut_rows, minimum_capacity, minimum_cuts, certificate).
View answer and explanation
([[[s], [[[s, a0], 18], [[s, a1], 19]], 37], [[a0, s], [[[a0, t], 18], [[s, a1], 19]], 37], [[a1, s], [[[a1, t], 19], [[s, a0], 18]], 37], [[a0, a1, s], [[[a0, t], 18], [[a1, t], 19]], 37]], 37, [[s], [a0, s], [a1, s], [a0, a1, s]], minimum_cut_is_selected_from_the_complete_cut_family)
Retain all cut rows, the least exact capacity and every minimizing source-side set. A single small cut without complete comparison does not establish a minimum. The finite network certificate does not prove the general theorem. The exact result, retaining the requested order, is ([[[s], [[[s, a0], 18], [[s, a1], 19]], 37], [[a0, s], [[[a0, t], 18], [[s, a1], 19]], 37], [[a1, s], [[[a1, t], 19], [[s, a0], 18]], 37], [[a0, a1, s], [[[a0, t], 18], [[a1, t], 19]], 37]], 37, [[s], [a0, s], [a1, s], [a0, a1, s]], minimum_cut_is_selected_from_the_complete_cut_family).
Reviewed step-by-step derivation
Max-flow min-cut theorem · qf-mod46-mfmc-minimum-cut-audit · candidate
68. The Minimum Cut Is Selected from the Complete Cut Family
Use complete directed vertices=[s, a0, a1, a2, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t], [s, a2], [a2, t]], source=s, sink=t, aligned nonnegative exact capacities=[19, 19, 20, 20, 21, 21]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (cut_rows, minimum_capacity, minimum_cuts, certificate).
View answer and explanation
([[[s], [[[s, a0], 19], [[s, a1], 20], [[s, a2], 21]], 60], [[a0, s], [[[a0, t], 19], [[s, a1], 20], [[s, a2], 21]], 60], [[a1, s], [[[a1, t], 20], [[s, a0], 19], [[s, a2], 21]], 60], [[a0, a1, s], [[[a0, t], 19], [[a1, t], 20], [[s, a2], 21]], 60], [[a2, s], [[[a2, t], 21], [[s, a0], 19], [[s, a1], 20]], 60], [[a0, a2, s], [[[a0, t], 19], [[a2, t], 21], [[s, a1], 20]], 60], [[a1, a2, s], [[[a1, t], 20], [[a2, t], 21], [[s, a0], 19]], 60], [[a0, a1, a2, s], [[[a0, t], 19], [[a1, t], 20], [[a2, t], 21]], 60]], 60, [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s]], minimum_cut_is_selected_from_the_complete_cut_family)
Retain all cut rows, the least exact capacity and every minimizing source-side set. A single small cut without complete comparison does not establish a minimum. The finite network certificate does not prove the general theorem. The exact result, retaining the requested order, is ([[[s], [[[s, a0], 19], [[s, a1], 20], [[s, a2], 21]], 60], [[a0, s], [[[a0, t], 19], [[s, a1], 20], [[s, a2], 21]], 60], [[a1, s], [[[a1, t], 20], [[s, a0], 19], [[s, a2], 21]], 60], [[a0, a1, s], [[[a0, t], 19], [[a1, t], 20], [[s, a2], 21]], 60], [[a2, s], [[[a2, t], 21], [[s, a0], 19], [[s, a1], 20]], 60], [[a0, a2, s], [[[a0, t], 19], [[a2, t], 21], [[s, a1], 20]], 60], [[a1, a2, s], [[[a1, t], 20], [[a2, t], 21], [[s, a0], 19]], 60], [[a0, a1, a2, s], [[[a0, t], 19], [[a1, t], 20], [[a2, t], 21]], 60]], 60, [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s]], minimum_cut_is_selected_from_the_complete_cut_family).
Reviewed step-by-step derivation
Max-flow min-cut theorem · qf-mod46-mfmc-minimum-cut-audit · candidate
69. The Minimum Cut Is Selected from the Complete Cut Family
Use complete directed vertices=[s, a0, a1, a2, a3, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t], [s, a2], [a2, t], [s, a3], [a3, t], [a0, s]], source=s, sink=t, aligned nonnegative exact capacities=[20, 20, 21, 21, 22, 22, 23, 23, 25]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (cut_rows, minimum_capacity, minimum_cuts, certificate).
View answer and explanation
([[[s], [[[s, a0], 20], [[s, a1], 21], [[s, a2], 22], [[s, a3], 23]], 86], [[a0, s], [[[a0, t], 20], [[s, a1], 21], [[s, a2], 22], [[s, a3], 23]], 86], [[a1, s], [[[a1, t], 21], [[s, a0], 20], [[s, a2], 22], [[s, a3], 23]], 86], [[a0, a1, s], [[[a0, t], 20], [[a1, t], 21], [[s, a2], 22], [[s, a3], 23]], 86], [[a2, s], [[[a2, t], 22], [[s, a0], 20], [[s, a1], 21], [[s, a3], 23]], 86], [[a0, a2, s], [[[a0, t], 20], [[a2, t], 22], [[s, a1], 21], [[s, a3], 23]], 86], [[a1, a2, s], [[[a1, t], 21], [[a2, t], 22], [[s, a0], 20], [[s, a3], 23]], 86], [[a0, a1, a2, s], [[[a0, t], 20], [[a1, t], 21], [[a2, t], 22], [[s, a3], 23]], 86], [[a3, s], [[[a3, t], 23], [[s, a0], 20], [[s, a1], 21], [[s, a2], 22]], 86], [[a0, a3, s], [[[a0, t], 20], [[a3, t], 23], [[s, a1], 21], [[s, a2], 22]], 86], [[a1, a3, s], [[[a1, t], 21], [[a3, t], 23], [[s, a0], 20], [[s, a2], 22]], 86], [[a0, a1, a3, s], [[[a0, t], 20], [[a1, t], 21], [[a3, t], 23], [[s, a2], 22]], 86], [[a2, a3, s], [[[a2, t], 22], [[a3, t], 23], [[s, a0], 20], [[s, a1], 21]], 86], [[a0, a2, a3, s], [[[a0, t], 20], [[a2, t], 22], [[a3, t], 23], [[s, a1], 21]], 86], [[a1, a2, a3, s], [[[a1, t], 21], [[a2, t], 22], [[a3, t], 23], [[s, a0], 20]], 86], [[a0, a1, a2, a3, s], [[[a0, t], 20], [[a1, t], 21], [[a2, t], 22], [[a3, t], 23]], 86]], 86, [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s], [a3, s], [a0, a3, s], [a1, a3, s], [a0, a1, a3, s], [a2, a3, s], [a0, a2, a3, s], [a1, a2, a3, s], [a0, a1, a2, a3, s]], minimum_cut_is_selected_from_the_complete_cut_family)
Retain all cut rows, the least exact capacity and every minimizing source-side set. A single small cut without complete comparison does not establish a minimum. The finite network certificate does not prove the general theorem. The exact result, retaining the requested order, is ([[[s], [[[s, a0], 20], [[s, a1], 21], [[s, a2], 22], [[s, a3], 23]], 86], [[a0, s], [[[a0, t], 20], [[s, a1], 21], [[s, a2], 22], [[s, a3], 23]], 86], [[a1, s], [[[a1, t], 21], [[s, a0], 20], [[s, a2], 22], [[s, a3], 23]], 86], [[a0, a1, s], [[[a0, t], 20], [[a1, t], 21], [[s, a2], 22], [[s, a3], 23]], 86], [[a2, s], [[[a2, t], 22], [[s, a0], 20], [[s, a1], 21], [[s, a3], 23]], 86], [[a0, a2, s], [[[a0, t], 20], [[a2, t], 22], [[s, a1], 21], [[s, a3], 23]], 86], [[a1, a2, s], [[[a1, t], 21], [[a2, t], 22], [[s, a0], 20], [[s, a3], 23]], 86], [[a0, a1, a2, s], [[[a0, t], 20], [[a1, t], 21], [[a2, t], 22], [[s, a3], 23]], 86], [[a3, s], [[[a3, t], 23], [[s, a0], 20], [[s, a1], 21], [[s, a2], 22]], 86], [[a0, a3, s], [[[a0, t], 20], [[a3, t], 23], [[s, a1], 21], [[s, a2], 22]], 86], [[a1, a3, s], [[[a1, t], 21], [[a3, t], 23], [[s, a0], 20], [[s, a2], 22]], 86], [[a0, a1, a3, s], [[[a0, t], 20], [[a1, t], 21], [[a3, t], 23], [[s, a2], 22]], 86], [[a2, a3, s], [[[a2, t], 22], [[a3, t], 23], [[s, a0], 20], [[s, a1], 21]], 86], [[a0, a2, a3, s], [[[a0, t], 20], [[a2, t], 22], [[a3, t], 23], [[s, a1], 21]], 86], [[a1, a2, a3, s], [[[a1, t], 21], [[a2, t], 22], [[a3, t], 23], [[s, a0], 20]], 86], [[a0, a1, a2, a3, s], [[[a0, t], 20], [[a1, t], 21], [[a2, t], 22], [[a3, t], 23]], 86]], 86, [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s], [a3, s], [a0, a3, s], [a1, a3, s], [a0, a1, a3, s], [a2, a3, s], [a0, a2, a3, s], [a1, a2, a3, s], [a0, a1, a2, a3, s]], minimum_cut_is_selected_from_the_complete_cut_family).
Reviewed step-by-step derivation
Max-flow min-cut theorem · qf-mod46-mfmc-minimum-cut-audit · candidate
70. The Minimum Cut Is Selected from the Complete Cut Family
Use complete directed vertices=[s, a0, a1, a2, a3, a4, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t], [s, a2], [a2, t], [s, a3], [a3, t], [s, a4], [a4, t]], source=s, sink=t, aligned nonnegative exact capacities=[21, 21, 22, 22, 23, 23, 24, 24, 25, 25]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (cut_rows, minimum_capacity, minimum_cuts, certificate).
View answer and explanation
([[[s], [[[s, a0], 21], [[s, a1], 22], [[s, a2], 23], [[s, a3], 24], [[s, a4], 25]], 115], [[a0, s], [[[a0, t], 21], [[s, a1], 22], [[s, a2], 23], [[s, a3], 24], [[s, a4], 25]], 115], [[a1, s], [[[a1, t], 22], [[s, a0], 21], [[s, a2], 23], [[s, a3], 24], [[s, a4], 25]], 115], [[a0, a1, s], [[[a0, t], 21], [[a1, t], 22], [[s, a2], 23], [[s, a3], 24], [[s, a4], 25]], 115], [[a2, s], [[[a2, t], 23], [[s, a0], 21], [[s, a1], 22], [[s, a3], 24], [[s, a4], 25]], 115], [[a0, a2, s], [[[a0, t], 21], [[a2, t], 23], [[s, a1], 22], [[s, a3], 24], [[s, a4], 25]], 115], [[a1, a2, s], [[[a1, t], 22], [[a2, t], 23], [[s, a0], 21], [[s, a3], 24], [[s, a4], 25]], 115], [[a0, a1, a2, s], [[[a0, t], 21], [[a1, t], 22], [[a2, t], 23], [[s, a3], 24], [[s, a4], 25]], 115], [[a3, s], [[[a3, t], 24], [[s, a0], 21], [[s, a1], 22], [[s, a2], 23], [[s, a4], 25]], 115], [[a0, a3, s], [[[a0, t], 21], [[a3, t], 24], [[s, a1], 22], [[s, a2], 23], [[s, a4], 25]], 115], [[a1, a3, s], [[[a1, t], 22], [[a3, t], 24], [[s, a0], 21], [[s, a2], 23], [[s, a4], 25]], 115], [[a0, a1, a3, s], [[[a0, t], 21], [[a1, t], 22], [[a3, t], 24], [[s, a2], 23], [[s, a4], 25]], 115], [[a2, a3, s], [[[a2, t], 23], [[a3, t], 24], [[s, a0], 21], [[s, a1], 22], [[s, a4], 25]], 115], [[a0, a2, a3, s], [[[a0, t], 21], [[a2, t], 23], [[a3, t], 24], [[s, a1], 22], [[s, a4], 25]], 115], [[a1, a2, a3, s], [[[a1, t], 22], [[a2, t], 23], [[a3, t], 24], [[s, a0], 21], [[s, a4], 25]], 115], [[a0, a1, a2, a3, s], [[[a0, t], 21], [[a1, t], 22], [[a2, t], 23], [[a3, t], 24], [[s, a4], 25]], 115], [[a4, s], [[[a4, t], 25], [[s, a0], 21], [[s, a1], 22], [[s, a2], 23], [[s, a3], 24]], 115], [[a0, a4, s], [[[a0, t], 21], [[a4, t], 25], [[s, a1], 22], [[s, a2], 23], [[s, a3], 24]], 115], [[a1, a4, s], [[[a1, t], 22], [[a4, t], 25], [[s, a0], 21], [[s, a2], 23], [[s, a3], 24]], 115], [[a0, a1, a4, s], [[[a0, t], 21], [[a1, t], 22], [[a4, t], 25], [[s, a2], 23], [[s, a3], 24]], 115], [[a2, a4, s], [[[a2, t], 23], [[a4, t], 25], [[s, a0], 21], [[s, a1], 22], [[s, a3], 24]], 115], [[a0, a2, a4, s], [[[a0, t], 21], [[a2, t], 23], [[a4, t], 25], [[s, a1], 22], [[s, a3], 24]], 115], [[a1, a2, a4, s], [[[a1, t], 22], [[a2, t], 23], [[a4, t], 25], [[s, a0], 21], [[s, a3], 24]], 115], [[a0, a1, a2, a4, s], [[[a0, t], 21], [[a1, t], 22], [[a2, t], 23], [[a4, t], 25], [[s, a3], 24]], 115], [[a3, a4, s], [[[a3, t], 24], [[a4, t], 25], [[s, a0], 21], [[s, a1], 22], [[s, a2], 23]], 115], [[a0, a3, a4, s], [[[a0, t], 21], [[a3, t], 24], [[a4, t], 25], [[s, a1], 22], [[s, a2], 23]], 115], [[a1, a3, a4, s], [[[a1, t], 22], [[a3, t], 24], [[a4, t], 25], [[s, a0], 21], [[s, a2], 23]], 115], [[a0, a1, a3, a4, s], [[[a0, t], 21], [[a1, t], 22], [[a3, t], 24], [[a4, t], 25], [[s, a2], 23]], 115], [[a2, a3, a4, s], [[[a2, t], 23], [[a3, t], 24], [[a4, t], 25], [[s, a0], 21], [[s, a1], 22]], 115], [[a0, a2, a3, a4, s], [[[a0, t], 21], [[a2, t], 23], [[a3, t], 24], [[a4, t], 25], [[s, a1], 22]], 115], [[a1, a2, a3, a4, s], [[[a1, t], 22], [[a2, t], 23], [[a3, t], 24], [[a4, t], 25], [[s, a0], 21]], 115], [[a0, a1, a2, a3, a4, s], [[[a0, t], 21], [[a1, t], 22], [[a2, t], 23], [[a3, t], 24], [[a4, t], 25]], 115]], 115, [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s], [a3, s], [a0, a3, s], [a1, a3, s], [a0, a1, a3, s], [a2, a3, s], [a0, a2, a3, s], [a1, a2, a3, s], [a0, a1, a2, a3, s], [a4, s], [a0, a4, s], [a1, a4, s], [a0, a1, a4, s], [a2, a4, s], [a0, a2, a4, s], [a1, a2, a4, s], [a0, a1, a2, a4, s], [a3, a4, s], [a0, a3, a4, s], [a1, a3, a4, s], [a0, a1, a3, a4, s], [a2, a3, a4, s], [a0, a2, a3, a4, s], [a1, a2, a3, a4, s], [a0, a1, a2, a3, a4, s]], minimum_cut_is_selected_from_the_complete_cut_family)
Retain all cut rows, the least exact capacity and every minimizing source-side set. A single small cut without complete comparison does not establish a minimum. The finite network certificate does not prove the general theorem. The exact result, retaining the requested order, is ([[[s], [[[s, a0], 21], [[s, a1], 22], [[s, a2], 23], [[s, a3], 24], [[s, a4], 25]], 115], [[a0, s], [[[a0, t], 21], [[s, a1], 22], [[s, a2], 23], [[s, a3], 24], [[s, a4], 25]], 115], [[a1, s], [[[a1, t], 22], [[s, a0], 21], [[s, a2], 23], [[s, a3], 24], [[s, a4], 25]], 115], [[a0, a1, s], [[[a0, t], 21], [[a1, t], 22], [[s, a2], 23], [[s, a3], 24], [[s, a4], 25]], 115], [[a2, s], [[[a2, t], 23], [[s, a0], 21], [[s, a1], 22], [[s, a3], 24], [[s, a4], 25]], 115], [[a0, a2, s], [[[a0, t], 21], [[a2, t], 23], [[s, a1], 22], [[s, a3], 24], [[s, a4], 25]], 115], [[a1, a2, s], [[[a1, t], 22], [[a2, t], 23], [[s, a0], 21], [[s, a3], 24], [[s, a4], 25]], 115], [[a0, a1, a2, s], [[[a0, t], 21], [[a1, t], 22], [[a2, t], 23], [[s, a3], 24], [[s, a4], 25]], 115], [[a3, s], [[[a3, t], 24], [[s, a0], 21], [[s, a1], 22], [[s, a2], 23], [[s, a4], 25]], 115], [[a0, a3, s], [[[a0, t], 21], [[a3, t], 24], [[s, a1], 22], [[s, a2], 23], [[s, a4], 25]], 115], [[a1, a3, s], [[[a1, t], 22], [[a3, t], 24], [[s, a0], 21], [[s, a2], 23], [[s, a4], 25]], 115], [[a0, a1, a3, s], [[[a0, t], 21], [[a1, t], 22], [[a3, t], 24], [[s, a2], 23], [[s, a4], 25]], 115], [[a2, a3, s], [[[a2, t], 23], [[a3, t], 24], [[s, a0], 21], [[s, a1], 22], [[s, a4], 25]], 115], [[a0, a2, a3, s], [[[a0, t], 21], [[a2, t], 23], [[a3, t], 24], [[s, a1], 22], [[s, a4], 25]], 115], [[a1, a2, a3, s], [[[a1, t], 22], [[a2, t], 23], [[a3, t], 24], [[s, a0], 21], [[s, a4], 25]], 115], [[a0, a1, a2, a3, s], [[[a0, t], 21], [[a1, t], 22], [[a2, t], 23], [[a3, t], 24], [[s, a4], 25]], 115], [[a4, s], [[[a4, t], 25], [[s, a0], 21], [[s, a1], 22], [[s, a2], 23], [[s, a3], 24]], 115], [[a0, a4, s], [[[a0, t], 21], [[a4, t], 25], [[s, a1], 22], [[s, a2], 23], [[s, a3], 24]], 115], [[a1, a4, s], [[[a1, t], 22], [[a4, t], 25], [[s, a0], 21], [[s, a2], 23], [[s, a3], 24]], 115], [[a0, a1, a4, s], [[[a0, t], 21], [[a1, t], 22], [[a4, t], 25], [[s, a2], 23], [[s, a3], 24]], 115], [[a2, a4, s], [[[a2, t], 23], [[a4, t], 25], [[s, a0], 21], [[s, a1], 22], [[s, a3], 24]], 115], [[a0, a2, a4, s], [[[a0, t], 21], [[a2, t], 23], [[a4, t], 25], [[s, a1], 22], [[s, a3], 24]], 115], [[a1, a2, a4, s], [[[a1, t], 22], [[a2, t], 23], [[a4, t], 25], [[s, a0], 21], [[s, a3], 24]], 115], [[a0, a1, a2, a4, s], [[[a0, t], 21], [[a1, t], 22], [[a2, t], 23], [[a4, t], 25], [[s, a3], 24]], 115], [[a3, a4, s], [[[a3, t], 24], [[a4, t], 25], [[s, a0], 21], [[s, a1], 22], [[s, a2], 23]], 115], [[a0, a3, a4, s], [[[a0, t], 21], [[a3, t], 24], [[a4, t], 25], [[s, a1], 22], [[s, a2], 23]], 115], [[a1, a3, a4, s], [[[a1, t], 22], [[a3, t], 24], [[a4, t], 25], [[s, a0], 21], [[s, a2], 23]], 115], [[a0, a1, a3, a4, s], [[[a0, t], 21], [[a1, t], 22], [[a3, t], 24], [[a4, t], 25], [[s, a2], 23]], 115], [[a2, a3, a4, s], [[[a2, t], 23], [[a3, t], 24], [[a4, t], 25], [[s, a0], 21], [[s, a1], 22]], 115], [[a0, a2, a3, a4, s], [[[a0, t], 21], [[a2, t], 23], [[a3, t], 24], [[a4, t], 25], [[s, a1], 22]], 115], [[a1, a2, a3, a4, s], [[[a1, t], 22], [[a2, t], 23], [[a3, t], 24], [[a4, t], 25], [[s, a0], 21]], 115], [[a0, a1, a2, a3, a4, s], [[[a0, t], 21], [[a1, t], 22], [[a2, t], 23], [[a3, t], 24], [[a4, t], 25]], 115]], 115, [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s], [a3, s], [a0, a3, s], [a1, a3, s], [a0, a1, a3, s], [a2, a3, s], [a0, a2, a3, s], [a1, a2, a3, s], [a0, a1, a2, a3, s], [a4, s], [a0, a4, s], [a1, a4, s], [a0, a1, a4, s], [a2, a4, s], [a0, a2, a4, s], [a1, a2, a4, s], [a0, a1, a2, a4, s], [a3, a4, s], [a0, a3, a4, s], [a1, a3, a4, s], [a0, a1, a3, a4, s], [a2, a3, a4, s], [a0, a2, a3, a4, s], [a1, a2, a3, a4, s], [a0, a1, a2, a3, a4, s]], minimum_cut_is_selected_from_the_complete_cut_family).
Reviewed step-by-step derivation
Max-flow min-cut theorem · qf-mod46-mfmc-minimum-cut-audit · candidate
71. The Minimum Cut Is Selected from the Complete Cut Family
Use complete directed vertices=[s, a0, a1, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t]], source=s, sink=t, aligned nonnegative exact capacities=[22, 22, 23, 23]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (cut_rows, minimum_capacity, minimum_cuts, certificate).
View answer and explanation
([[[s], [[[s, a0], 22], [[s, a1], 23]], 45], [[a0, s], [[[a0, t], 22], [[s, a1], 23]], 45], [[a1, s], [[[a1, t], 23], [[s, a0], 22]], 45], [[a0, a1, s], [[[a0, t], 22], [[a1, t], 23]], 45]], 45, [[s], [a0, s], [a1, s], [a0, a1, s]], minimum_cut_is_selected_from_the_complete_cut_family)
Retain all cut rows, the least exact capacity and every minimizing source-side set. A single small cut without complete comparison does not establish a minimum. The finite network certificate does not prove the general theorem. The exact result, retaining the requested order, is ([[[s], [[[s, a0], 22], [[s, a1], 23]], 45], [[a0, s], [[[a0, t], 22], [[s, a1], 23]], 45], [[a1, s], [[[a1, t], 23], [[s, a0], 22]], 45], [[a0, a1, s], [[[a0, t], 22], [[a1, t], 23]], 45]], 45, [[s], [a0, s], [a1, s], [a0, a1, s]], minimum_cut_is_selected_from_the_complete_cut_family).
Reviewed step-by-step derivation
Max-flow min-cut theorem · qf-mod46-mfmc-minimum-cut-audit · candidate
72. The Minimum Cut Is Selected from the Complete Cut Family
Use complete directed vertices=[s, a0, a1, a2, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t], [s, a2], [a2, t], [a0, s]], source=s, sink=t, aligned nonnegative exact capacities=[23, 23, 24, 24, 25, 25, 28]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (cut_rows, minimum_capacity, minimum_cuts, certificate).
View answer and explanation
([[[s], [[[s, a0], 23], [[s, a1], 24], [[s, a2], 25]], 72], [[a0, s], [[[a0, t], 23], [[s, a1], 24], [[s, a2], 25]], 72], [[a1, s], [[[a1, t], 24], [[s, a0], 23], [[s, a2], 25]], 72], [[a0, a1, s], [[[a0, t], 23], [[a1, t], 24], [[s, a2], 25]], 72], [[a2, s], [[[a2, t], 25], [[s, a0], 23], [[s, a1], 24]], 72], [[a0, a2, s], [[[a0, t], 23], [[a2, t], 25], [[s, a1], 24]], 72], [[a1, a2, s], [[[a1, t], 24], [[a2, t], 25], [[s, a0], 23]], 72], [[a0, a1, a2, s], [[[a0, t], 23], [[a1, t], 24], [[a2, t], 25]], 72]], 72, [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s]], minimum_cut_is_selected_from_the_complete_cut_family)
Retain all cut rows, the least exact capacity and every minimizing source-side set. A single small cut without complete comparison does not establish a minimum. The finite network certificate does not prove the general theorem. The exact result, retaining the requested order, is ([[[s], [[[s, a0], 23], [[s, a1], 24], [[s, a2], 25]], 72], [[a0, s], [[[a0, t], 23], [[s, a1], 24], [[s, a2], 25]], 72], [[a1, s], [[[a1, t], 24], [[s, a0], 23], [[s, a2], 25]], 72], [[a0, a1, s], [[[a0, t], 23], [[a1, t], 24], [[s, a2], 25]], 72], [[a2, s], [[[a2, t], 25], [[s, a0], 23], [[s, a1], 24]], 72], [[a0, a2, s], [[[a0, t], 23], [[a2, t], 25], [[s, a1], 24]], 72], [[a1, a2, s], [[[a1, t], 24], [[a2, t], 25], [[s, a0], 23]], 72], [[a0, a1, a2, s], [[[a0, t], 23], [[a1, t], 24], [[a2, t], 25]], 72]], 72, [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s]], minimum_cut_is_selected_from_the_complete_cut_family).
Reviewed step-by-step derivation
Max-flow min-cut theorem · qf-mod46-mfmc-minimum-cut-audit · candidate
73. The Minimum Cut Is Selected from the Complete Cut Family
Use complete directed vertices=[s, a0, a1, a2, a3, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t], [s, a2], [a2, t], [s, a3], [a3, t]], source=s, sink=t, aligned nonnegative exact capacities=[24, 24, 25, 25, 26, 26, 27, 27]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (cut_rows, minimum_capacity, minimum_cuts, certificate).
View answer and explanation
([[[s], [[[s, a0], 24], [[s, a1], 25], [[s, a2], 26], [[s, a3], 27]], 102], [[a0, s], [[[a0, t], 24], [[s, a1], 25], [[s, a2], 26], [[s, a3], 27]], 102], [[a1, s], [[[a1, t], 25], [[s, a0], 24], [[s, a2], 26], [[s, a3], 27]], 102], [[a0, a1, s], [[[a0, t], 24], [[a1, t], 25], [[s, a2], 26], [[s, a3], 27]], 102], [[a2, s], [[[a2, t], 26], [[s, a0], 24], [[s, a1], 25], [[s, a3], 27]], 102], [[a0, a2, s], [[[a0, t], 24], [[a2, t], 26], [[s, a1], 25], [[s, a3], 27]], 102], [[a1, a2, s], [[[a1, t], 25], [[a2, t], 26], [[s, a0], 24], [[s, a3], 27]], 102], [[a0, a1, a2, s], [[[a0, t], 24], [[a1, t], 25], [[a2, t], 26], [[s, a3], 27]], 102], [[a3, s], [[[a3, t], 27], [[s, a0], 24], [[s, a1], 25], [[s, a2], 26]], 102], [[a0, a3, s], [[[a0, t], 24], [[a3, t], 27], [[s, a1], 25], [[s, a2], 26]], 102], [[a1, a3, s], [[[a1, t], 25], [[a3, t], 27], [[s, a0], 24], [[s, a2], 26]], 102], [[a0, a1, a3, s], [[[a0, t], 24], [[a1, t], 25], [[a3, t], 27], [[s, a2], 26]], 102], [[a2, a3, s], [[[a2, t], 26], [[a3, t], 27], [[s, a0], 24], [[s, a1], 25]], 102], [[a0, a2, a3, s], [[[a0, t], 24], [[a2, t], 26], [[a3, t], 27], [[s, a1], 25]], 102], [[a1, a2, a3, s], [[[a1, t], 25], [[a2, t], 26], [[a3, t], 27], [[s, a0], 24]], 102], [[a0, a1, a2, a3, s], [[[a0, t], 24], [[a1, t], 25], [[a2, t], 26], [[a3, t], 27]], 102]], 102, [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s], [a3, s], [a0, a3, s], [a1, a3, s], [a0, a1, a3, s], [a2, a3, s], [a0, a2, a3, s], [a1, a2, a3, s], [a0, a1, a2, a3, s]], minimum_cut_is_selected_from_the_complete_cut_family)
Retain all cut rows, the least exact capacity and every minimizing source-side set. A single small cut without complete comparison does not establish a minimum. The finite network certificate does not prove the general theorem. The exact result, retaining the requested order, is ([[[s], [[[s, a0], 24], [[s, a1], 25], [[s, a2], 26], [[s, a3], 27]], 102], [[a0, s], [[[a0, t], 24], [[s, a1], 25], [[s, a2], 26], [[s, a3], 27]], 102], [[a1, s], [[[a1, t], 25], [[s, a0], 24], [[s, a2], 26], [[s, a3], 27]], 102], [[a0, a1, s], [[[a0, t], 24], [[a1, t], 25], [[s, a2], 26], [[s, a3], 27]], 102], [[a2, s], [[[a2, t], 26], [[s, a0], 24], [[s, a1], 25], [[s, a3], 27]], 102], [[a0, a2, s], [[[a0, t], 24], [[a2, t], 26], [[s, a1], 25], [[s, a3], 27]], 102], [[a1, a2, s], [[[a1, t], 25], [[a2, t], 26], [[s, a0], 24], [[s, a3], 27]], 102], [[a0, a1, a2, s], [[[a0, t], 24], [[a1, t], 25], [[a2, t], 26], [[s, a3], 27]], 102], [[a3, s], [[[a3, t], 27], [[s, a0], 24], [[s, a1], 25], [[s, a2], 26]], 102], [[a0, a3, s], [[[a0, t], 24], [[a3, t], 27], [[s, a1], 25], [[s, a2], 26]], 102], [[a1, a3, s], [[[a1, t], 25], [[a3, t], 27], [[s, a0], 24], [[s, a2], 26]], 102], [[a0, a1, a3, s], [[[a0, t], 24], [[a1, t], 25], [[a3, t], 27], [[s, a2], 26]], 102], [[a2, a3, s], [[[a2, t], 26], [[a3, t], 27], [[s, a0], 24], [[s, a1], 25]], 102], [[a0, a2, a3, s], [[[a0, t], 24], [[a2, t], 26], [[a3, t], 27], [[s, a1], 25]], 102], [[a1, a2, a3, s], [[[a1, t], 25], [[a2, t], 26], [[a3, t], 27], [[s, a0], 24]], 102], [[a0, a1, a2, a3, s], [[[a0, t], 24], [[a1, t], 25], [[a2, t], 26], [[a3, t], 27]], 102]], 102, [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s], [a3, s], [a0, a3, s], [a1, a3, s], [a0, a1, a3, s], [a2, a3, s], [a0, a2, a3, s], [a1, a2, a3, s], [a0, a1, a2, a3, s]], minimum_cut_is_selected_from_the_complete_cut_family).
Reviewed step-by-step derivation
Max-flow min-cut theorem · qf-mod46-mfmc-minimum-cut-audit · candidate
74. The Minimum Cut Is Selected from the Complete Cut Family
Use complete directed vertices=[s, a0, a1, a2, a3, a4, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t], [s, a2], [a2, t], [s, a3], [a3, t], [s, a4], [a4, t]], source=s, sink=t, aligned nonnegative exact capacities=[25, 25, 26, 26, 27, 27, 28, 28, 29, 29]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (cut_rows, minimum_capacity, minimum_cuts, certificate).
View answer and explanation
([[[s], [[[s, a0], 25], [[s, a1], 26], [[s, a2], 27], [[s, a3], 28], [[s, a4], 29]], 135], [[a0, s], [[[a0, t], 25], [[s, a1], 26], [[s, a2], 27], [[s, a3], 28], [[s, a4], 29]], 135], [[a1, s], [[[a1, t], 26], [[s, a0], 25], [[s, a2], 27], [[s, a3], 28], [[s, a4], 29]], 135], [[a0, a1, s], [[[a0, t], 25], [[a1, t], 26], [[s, a2], 27], [[s, a3], 28], [[s, a4], 29]], 135], [[a2, s], [[[a2, t], 27], [[s, a0], 25], [[s, a1], 26], [[s, a3], 28], [[s, a4], 29]], 135], [[a0, a2, s], [[[a0, t], 25], [[a2, t], 27], [[s, a1], 26], [[s, a3], 28], [[s, a4], 29]], 135], [[a1, a2, s], [[[a1, t], 26], [[a2, t], 27], [[s, a0], 25], [[s, a3], 28], [[s, a4], 29]], 135], [[a0, a1, a2, s], [[[a0, t], 25], [[a1, t], 26], [[a2, t], 27], [[s, a3], 28], [[s, a4], 29]], 135], [[a3, s], [[[a3, t], 28], [[s, a0], 25], [[s, a1], 26], [[s, a2], 27], [[s, a4], 29]], 135], [[a0, a3, s], [[[a0, t], 25], [[a3, t], 28], [[s, a1], 26], [[s, a2], 27], [[s, a4], 29]], 135], [[a1, a3, s], [[[a1, t], 26], [[a3, t], 28], [[s, a0], 25], [[s, a2], 27], [[s, a4], 29]], 135], [[a0, a1, a3, s], [[[a0, t], 25], [[a1, t], 26], [[a3, t], 28], [[s, a2], 27], [[s, a4], 29]], 135], [[a2, a3, s], [[[a2, t], 27], [[a3, t], 28], [[s, a0], 25], [[s, a1], 26], [[s, a4], 29]], 135], [[a0, a2, a3, s], [[[a0, t], 25], [[a2, t], 27], [[a3, t], 28], [[s, a1], 26], [[s, a4], 29]], 135], [[a1, a2, a3, s], [[[a1, t], 26], [[a2, t], 27], [[a3, t], 28], [[s, a0], 25], [[s, a4], 29]], 135], [[a0, a1, a2, a3, s], [[[a0, t], 25], [[a1, t], 26], [[a2, t], 27], [[a3, t], 28], [[s, a4], 29]], 135], [[a4, s], [[[a4, t], 29], [[s, a0], 25], [[s, a1], 26], [[s, a2], 27], [[s, a3], 28]], 135], [[a0, a4, s], [[[a0, t], 25], [[a4, t], 29], [[s, a1], 26], [[s, a2], 27], [[s, a3], 28]], 135], [[a1, a4, s], [[[a1, t], 26], [[a4, t], 29], [[s, a0], 25], [[s, a2], 27], [[s, a3], 28]], 135], [[a0, a1, a4, s], [[[a0, t], 25], [[a1, t], 26], [[a4, t], 29], [[s, a2], 27], [[s, a3], 28]], 135], [[a2, a4, s], [[[a2, t], 27], [[a4, t], 29], [[s, a0], 25], [[s, a1], 26], [[s, a3], 28]], 135], [[a0, a2, a4, s], [[[a0, t], 25], [[a2, t], 27], [[a4, t], 29], [[s, a1], 26], [[s, a3], 28]], 135], [[a1, a2, a4, s], [[[a1, t], 26], [[a2, t], 27], [[a4, t], 29], [[s, a0], 25], [[s, a3], 28]], 135], [[a0, a1, a2, a4, s], [[[a0, t], 25], [[a1, t], 26], [[a2, t], 27], [[a4, t], 29], [[s, a3], 28]], 135], [[a3, a4, s], [[[a3, t], 28], [[a4, t], 29], [[s, a0], 25], [[s, a1], 26], [[s, a2], 27]], 135], [[a0, a3, a4, s], [[[a0, t], 25], [[a3, t], 28], [[a4, t], 29], [[s, a1], 26], [[s, a2], 27]], 135], [[a1, a3, a4, s], [[[a1, t], 26], [[a3, t], 28], [[a4, t], 29], [[s, a0], 25], [[s, a2], 27]], 135], [[a0, a1, a3, a4, s], [[[a0, t], 25], [[a1, t], 26], [[a3, t], 28], [[a4, t], 29], [[s, a2], 27]], 135], [[a2, a3, a4, s], [[[a2, t], 27], [[a3, t], 28], [[a4, t], 29], [[s, a0], 25], [[s, a1], 26]], 135], [[a0, a2, a3, a4, s], [[[a0, t], 25], [[a2, t], 27], [[a3, t], 28], [[a4, t], 29], [[s, a1], 26]], 135], [[a1, a2, a3, a4, s], [[[a1, t], 26], [[a2, t], 27], [[a3, t], 28], [[a4, t], 29], [[s, a0], 25]], 135], [[a0, a1, a2, a3, a4, s], [[[a0, t], 25], [[a1, t], 26], [[a2, t], 27], [[a3, t], 28], [[a4, t], 29]], 135]], 135, [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s], [a3, s], [a0, a3, s], [a1, a3, s], [a0, a1, a3, s], [a2, a3, s], [a0, a2, a3, s], [a1, a2, a3, s], [a0, a1, a2, a3, s], [a4, s], [a0, a4, s], [a1, a4, s], [a0, a1, a4, s], [a2, a4, s], [a0, a2, a4, s], [a1, a2, a4, s], [a0, a1, a2, a4, s], [a3, a4, s], [a0, a3, a4, s], [a1, a3, a4, s], [a0, a1, a3, a4, s], [a2, a3, a4, s], [a0, a2, a3, a4, s], [a1, a2, a3, a4, s], [a0, a1, a2, a3, a4, s]], minimum_cut_is_selected_from_the_complete_cut_family)
Retain all cut rows, the least exact capacity and every minimizing source-side set. A single small cut without complete comparison does not establish a minimum. The finite network certificate does not prove the general theorem. The exact result, retaining the requested order, is ([[[s], [[[s, a0], 25], [[s, a1], 26], [[s, a2], 27], [[s, a3], 28], [[s, a4], 29]], 135], [[a0, s], [[[a0, t], 25], [[s, a1], 26], [[s, a2], 27], [[s, a3], 28], [[s, a4], 29]], 135], [[a1, s], [[[a1, t], 26], [[s, a0], 25], [[s, a2], 27], [[s, a3], 28], [[s, a4], 29]], 135], [[a0, a1, s], [[[a0, t], 25], [[a1, t], 26], [[s, a2], 27], [[s, a3], 28], [[s, a4], 29]], 135], [[a2, s], [[[a2, t], 27], [[s, a0], 25], [[s, a1], 26], [[s, a3], 28], [[s, a4], 29]], 135], [[a0, a2, s], [[[a0, t], 25], [[a2, t], 27], [[s, a1], 26], [[s, a3], 28], [[s, a4], 29]], 135], [[a1, a2, s], [[[a1, t], 26], [[a2, t], 27], [[s, a0], 25], [[s, a3], 28], [[s, a4], 29]], 135], [[a0, a1, a2, s], [[[a0, t], 25], [[a1, t], 26], [[a2, t], 27], [[s, a3], 28], [[s, a4], 29]], 135], [[a3, s], [[[a3, t], 28], [[s, a0], 25], [[s, a1], 26], [[s, a2], 27], [[s, a4], 29]], 135], [[a0, a3, s], [[[a0, t], 25], [[a3, t], 28], [[s, a1], 26], [[s, a2], 27], [[s, a4], 29]], 135], [[a1, a3, s], [[[a1, t], 26], [[a3, t], 28], [[s, a0], 25], [[s, a2], 27], [[s, a4], 29]], 135], [[a0, a1, a3, s], [[[a0, t], 25], [[a1, t], 26], [[a3, t], 28], [[s, a2], 27], [[s, a4], 29]], 135], [[a2, a3, s], [[[a2, t], 27], [[a3, t], 28], [[s, a0], 25], [[s, a1], 26], [[s, a4], 29]], 135], [[a0, a2, a3, s], [[[a0, t], 25], [[a2, t], 27], [[a3, t], 28], [[s, a1], 26], [[s, a4], 29]], 135], [[a1, a2, a3, s], [[[a1, t], 26], [[a2, t], 27], [[a3, t], 28], [[s, a0], 25], [[s, a4], 29]], 135], [[a0, a1, a2, a3, s], [[[a0, t], 25], [[a1, t], 26], [[a2, t], 27], [[a3, t], 28], [[s, a4], 29]], 135], [[a4, s], [[[a4, t], 29], [[s, a0], 25], [[s, a1], 26], [[s, a2], 27], [[s, a3], 28]], 135], [[a0, a4, s], [[[a0, t], 25], [[a4, t], 29], [[s, a1], 26], [[s, a2], 27], [[s, a3], 28]], 135], [[a1, a4, s], [[[a1, t], 26], [[a4, t], 29], [[s, a0], 25], [[s, a2], 27], [[s, a3], 28]], 135], [[a0, a1, a4, s], [[[a0, t], 25], [[a1, t], 26], [[a4, t], 29], [[s, a2], 27], [[s, a3], 28]], 135], [[a2, a4, s], [[[a2, t], 27], [[a4, t], 29], [[s, a0], 25], [[s, a1], 26], [[s, a3], 28]], 135], [[a0, a2, a4, s], [[[a0, t], 25], [[a2, t], 27], [[a4, t], 29], [[s, a1], 26], [[s, a3], 28]], 135], [[a1, a2, a4, s], [[[a1, t], 26], [[a2, t], 27], [[a4, t], 29], [[s, a0], 25], [[s, a3], 28]], 135], [[a0, a1, a2, a4, s], [[[a0, t], 25], [[a1, t], 26], [[a2, t], 27], [[a4, t], 29], [[s, a3], 28]], 135], [[a3, a4, s], [[[a3, t], 28], [[a4, t], 29], [[s, a0], 25], [[s, a1], 26], [[s, a2], 27]], 135], [[a0, a3, a4, s], [[[a0, t], 25], [[a3, t], 28], [[a4, t], 29], [[s, a1], 26], [[s, a2], 27]], 135], [[a1, a3, a4, s], [[[a1, t], 26], [[a3, t], 28], [[a4, t], 29], [[s, a0], 25], [[s, a2], 27]], 135], [[a0, a1, a3, a4, s], [[[a0, t], 25], [[a1, t], 26], [[a3, t], 28], [[a4, t], 29], [[s, a2], 27]], 135], [[a2, a3, a4, s], [[[a2, t], 27], [[a3, t], 28], [[a4, t], 29], [[s, a0], 25], [[s, a1], 26]], 135], [[a0, a2, a3, a4, s], [[[a0, t], 25], [[a2, t], 27], [[a3, t], 28], [[a4, t], 29], [[s, a1], 26]], 135], [[a1, a2, a3, a4, s], [[[a1, t], 26], [[a2, t], 27], [[a3, t], 28], [[a4, t], 29], [[s, a0], 25]], 135], [[a0, a1, a2, a3, a4, s], [[[a0, t], 25], [[a1, t], 26], [[a2, t], 27], [[a3, t], 28], [[a4, t], 29]], 135]], 135, [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s], [a3, s], [a0, a3, s], [a1, a3, s], [a0, a1, a3, s], [a2, a3, s], [a0, a2, a3, s], [a1, a2, a3, s], [a0, a1, a2, a3, s], [a4, s], [a0, a4, s], [a1, a4, s], [a0, a1, a4, s], [a2, a4, s], [a0, a2, a4, s], [a1, a2, a4, s], [a0, a1, a2, a4, s], [a3, a4, s], [a0, a3, a4, s], [a1, a3, a4, s], [a0, a1, a3, a4, s], [a2, a3, a4, s], [a0, a2, a3, a4, s], [a1, a2, a3, a4, s], [a0, a1, a2, a3, a4, s]], minimum_cut_is_selected_from_the_complete_cut_family).
Reviewed step-by-step derivation
Max-flow min-cut theorem · qf-mod46-mfmc-minimum-cut-audit · candidate
75. The Minimum Cut Is Selected from the Complete Cut Family
Use complete directed vertices=[s, a0, a1, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t], [a0, s]], source=s, sink=t, aligned nonnegative exact capacities=[26, 26, 27, 27, 31]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (cut_rows, minimum_capacity, minimum_cuts, certificate).
View answer and explanation
([[[s], [[[s, a0], 26], [[s, a1], 27]], 53], [[a0, s], [[[a0, t], 26], [[s, a1], 27]], 53], [[a1, s], [[[a1, t], 27], [[s, a0], 26]], 53], [[a0, a1, s], [[[a0, t], 26], [[a1, t], 27]], 53]], 53, [[s], [a0, s], [a1, s], [a0, a1, s]], minimum_cut_is_selected_from_the_complete_cut_family)
Retain all cut rows, the least exact capacity and every minimizing source-side set. A single small cut without complete comparison does not establish a minimum. The finite network certificate does not prove the general theorem. The exact result, retaining the requested order, is ([[[s], [[[s, a0], 26], [[s, a1], 27]], 53], [[a0, s], [[[a0, t], 26], [[s, a1], 27]], 53], [[a1, s], [[[a1, t], 27], [[s, a0], 26]], 53], [[a0, a1, s], [[[a0, t], 26], [[a1, t], 27]], 53]], 53, [[s], [a0, s], [a1, s], [a0, a1, s]], minimum_cut_is_selected_from_the_complete_cut_family).
Reviewed step-by-step derivation
Max-flow min-cut theorem · qf-mod46-mfmc-minimum-cut-audit · candidate
76. An Equal-Valued Feasible Flow and Cut Certify Both Optima
Use complete directed vertices=[s, a0, a1, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t], [a0, s]], source=s, sink=t, aligned nonnegative exact capacities=[2, 2, 3, 3, 7], candidate flows=[2, 2, 3, 3, 0]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (flow_value, minimum_capacity, equal_value_cuts, minimum_cuts, equal_values, maximum_flow_certificate, minimum_cut_certificate, integrality_or_algorithm_claim, certificate).
Equal-value certification first requires a feasible flow. The final No withholds integrality or a particular algorithm proof, not maximum-flow/minimum-cut certification.
View answer and explanation
(5, 5, [[s], [a0, s], [a1, s], [a0, a1, s]], [[s], [a0, s], [a1, s], [a0, a1, s]], Yes, Yes, Yes, No, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims)
First verify the full flow feasibility conditions. Compare its value with a cut capacity and retain all equal-valued/minimizing cuts. The universal flow≤cut bound makes both optimal. The final No withholds integrality and algorithm claims, not the optimality certificate. The exact result, retaining the requested order, is (5, 5, [[s], [a0, s], [a1, s], [a0, a1, s]], [[s], [a0, s], [a1, s], [a0, a1, s]], Yes, Yes, Yes, No, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims).
Reviewed step-by-step derivation
Max-flow min-cut theorem · qf-mod46-mfmc-equal-value-optimality · candidate
77. An Equal-Valued Feasible Flow and Cut Certify Both Optima
Use complete directed vertices=[s, a0, a1, a2, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t], [s, a2], [a2, t]], source=s, sink=t, aligned nonnegative exact capacities=[3, 3, 4, 4, 5, 5], candidate flows=[3, 3, 4, 4, 5, 5]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (flow_value, minimum_capacity, equal_value_cuts, minimum_cuts, equal_values, maximum_flow_certificate, minimum_cut_certificate, integrality_or_algorithm_claim, certificate).
Equal-value certification first requires a feasible flow. The final No withholds integrality or a particular algorithm proof, not maximum-flow/minimum-cut certification.
View answer and explanation
(12, 12, [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s]], [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s]], Yes, Yes, Yes, No, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims)
First verify the full flow feasibility conditions. Compare its value with a cut capacity and retain all equal-valued/minimizing cuts. The universal flow≤cut bound makes both optimal. The final No withholds integrality and algorithm claims, not the optimality certificate. The exact result, retaining the requested order, is (12, 12, [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s]], [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s]], Yes, Yes, Yes, No, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims).
Reviewed step-by-step derivation
Max-flow min-cut theorem · qf-mod46-mfmc-equal-value-optimality · candidate
78. An Equal-Valued Feasible Flow and Cut Certify Both Optima
Use complete directed vertices=[s, a0, a1, a2, a3, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t], [s, a2], [a2, t], [s, a3], [a3, t]], source=s, sink=t, aligned nonnegative exact capacities=[4, 4, 5, 5, 6, 6, 7, 7], candidate flows=[4, 4, 5, 5, 6, 6, 7, 7]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (flow_value, minimum_capacity, equal_value_cuts, minimum_cuts, equal_values, maximum_flow_certificate, minimum_cut_certificate, integrality_or_algorithm_claim, certificate).
Equal-value certification first requires a feasible flow. The final No withholds integrality or a particular algorithm proof, not maximum-flow/minimum-cut certification.
View answer and explanation
(22, 22, [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s], [a3, s], [a0, a3, s], [a1, a3, s], [a0, a1, a3, s], [a2, a3, s], [a0, a2, a3, s], [a1, a2, a3, s], [a0, a1, a2, a3, s]], [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s], [a3, s], [a0, a3, s], [a1, a3, s], [a0, a1, a3, s], [a2, a3, s], [a0, a2, a3, s], [a1, a2, a3, s], [a0, a1, a2, a3, s]], Yes, Yes, Yes, No, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims)
First verify the full flow feasibility conditions. Compare its value with a cut capacity and retain all equal-valued/minimizing cuts. The universal flow≤cut bound makes both optimal. The final No withholds integrality and algorithm claims, not the optimality certificate. The exact result, retaining the requested order, is (22, 22, [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s], [a3, s], [a0, a3, s], [a1, a3, s], [a0, a1, a3, s], [a2, a3, s], [a0, a2, a3, s], [a1, a2, a3, s], [a0, a1, a2, a3, s]], [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s], [a3, s], [a0, a3, s], [a1, a3, s], [a0, a1, a3, s], [a2, a3, s], [a0, a2, a3, s], [a1, a2, a3, s], [a0, a1, a2, a3, s]], Yes, Yes, Yes, No, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims).
Reviewed step-by-step derivation
Max-flow min-cut theorem · qf-mod46-mfmc-equal-value-optimality · candidate
79. An Equal-Valued Feasible Flow and Cut Certify Both Optima
Use complete directed vertices=[s, a0, a1, a2, a3, a4, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t], [s, a2], [a2, t], [s, a3], [a3, t], [s, a4], [a4, t], [a0, s]], source=s, sink=t, aligned nonnegative exact capacities=[5, 5, 6, 6, 7, 7, 8, 8, 9, 9, 10], candidate flows=[5, 5, 6, 6, 7, 7, 8, 8, 9, 9, 0]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (flow_value, minimum_capacity, equal_value_cuts, minimum_cuts, equal_values, maximum_flow_certificate, minimum_cut_certificate, integrality_or_algorithm_claim, certificate).
Equal-value certification first requires a feasible flow. The final No withholds integrality or a particular algorithm proof, not maximum-flow/minimum-cut certification.
View answer and explanation
(35, 35, [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s], [a3, s], [a0, a3, s], [a1, a3, s], [a0, a1, a3, s], [a2, a3, s], [a0, a2, a3, s], [a1, a2, a3, s], [a0, a1, a2, a3, s], [a4, s], [a0, a4, s], [a1, a4, s], [a0, a1, a4, s], [a2, a4, s], [a0, a2, a4, s], [a1, a2, a4, s], [a0, a1, a2, a4, s], [a3, a4, s], [a0, a3, a4, s], [a1, a3, a4, s], [a0, a1, a3, a4, s], [a2, a3, a4, s], [a0, a2, a3, a4, s], [a1, a2, a3, a4, s], [a0, a1, a2, a3, a4, s]], [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s], [a3, s], [a0, a3, s], [a1, a3, s], [a0, a1, a3, s], [a2, a3, s], [a0, a2, a3, s], [a1, a2, a3, s], [a0, a1, a2, a3, s], [a4, s], [a0, a4, s], [a1, a4, s], [a0, a1, a4, s], [a2, a4, s], [a0, a2, a4, s], [a1, a2, a4, s], [a0, a1, a2, a4, s], [a3, a4, s], [a0, a3, a4, s], [a1, a3, a4, s], [a0, a1, a3, a4, s], [a2, a3, a4, s], [a0, a2, a3, a4, s], [a1, a2, a3, a4, s], [a0, a1, a2, a3, a4, s]], Yes, Yes, Yes, No, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims)
First verify the full flow feasibility conditions. Compare its value with a cut capacity and retain all equal-valued/minimizing cuts. The universal flow≤cut bound makes both optimal. The final No withholds integrality and algorithm claims, not the optimality certificate. The exact result, retaining the requested order, is (35, 35, [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s], [a3, s], [a0, a3, s], [a1, a3, s], [a0, a1, a3, s], [a2, a3, s], [a0, a2, a3, s], [a1, a2, a3, s], [a0, a1, a2, a3, s], [a4, s], [a0, a4, s], [a1, a4, s], [a0, a1, a4, s], [a2, a4, s], [a0, a2, a4, s], [a1, a2, a4, s], [a0, a1, a2, a4, s], [a3, a4, s], [a0, a3, a4, s], [a1, a3, a4, s], [a0, a1, a3, a4, s], [a2, a3, a4, s], [a0, a2, a3, a4, s], [a1, a2, a3, a4, s], [a0, a1, a2, a3, a4, s]], [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s], [a3, s], [a0, a3, s], [a1, a3, s], [a0, a1, a3, s], [a2, a3, s], [a0, a2, a3, s], [a1, a2, a3, s], [a0, a1, a2, a3, s], [a4, s], [a0, a4, s], [a1, a4, s], [a0, a1, a4, s], [a2, a4, s], [a0, a2, a4, s], [a1, a2, a4, s], [a0, a1, a2, a4, s], [a3, a4, s], [a0, a3, a4, s], [a1, a3, a4, s], [a0, a1, a3, a4, s], [a2, a3, a4, s], [a0, a2, a3, a4, s], [a1, a2, a3, a4, s], [a0, a1, a2, a3, a4, s]], Yes, Yes, Yes, No, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims).
Reviewed step-by-step derivation
Max-flow min-cut theorem · qf-mod46-mfmc-equal-value-optimality · candidate
80. An Equal-Valued Feasible Flow and Cut Certify Both Optima
Use complete directed vertices=[s, a0, a1, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t]], source=s, sink=t, aligned nonnegative exact capacities=[6, 6, 7, 7], candidate flows=[6, 6, 7, 7]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (flow_value, minimum_capacity, equal_value_cuts, minimum_cuts, equal_values, maximum_flow_certificate, minimum_cut_certificate, integrality_or_algorithm_claim, certificate).
Equal-value certification first requires a feasible flow. The final No withholds integrality or a particular algorithm proof, not maximum-flow/minimum-cut certification.
View answer and explanation
(13, 13, [[s], [a0, s], [a1, s], [a0, a1, s]], [[s], [a0, s], [a1, s], [a0, a1, s]], Yes, Yes, Yes, No, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims)
First verify the full flow feasibility conditions. Compare its value with a cut capacity and retain all equal-valued/minimizing cuts. The universal flow≤cut bound makes both optimal. The final No withholds integrality and algorithm claims, not the optimality certificate. The exact result, retaining the requested order, is (13, 13, [[s], [a0, s], [a1, s], [a0, a1, s]], [[s], [a0, s], [a1, s], [a0, a1, s]], Yes, Yes, Yes, No, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims).
Reviewed step-by-step derivation
Max-flow min-cut theorem · qf-mod46-mfmc-equal-value-optimality · candidate
81. An Equal-Valued Feasible Flow and Cut Certify Both Optima
Use complete directed vertices=[s, a0, a1, a2, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t], [s, a2], [a2, t]], source=s, sink=t, aligned nonnegative exact capacities=[7, 7, 8, 8, 9, 9], candidate flows=[7, 7, 8, 8, 9, 9]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (flow_value, minimum_capacity, equal_value_cuts, minimum_cuts, equal_values, maximum_flow_certificate, minimum_cut_certificate, integrality_or_algorithm_claim, certificate).
Equal-value certification first requires a feasible flow. The final No withholds integrality or a particular algorithm proof, not maximum-flow/minimum-cut certification.
View answer and explanation
(24, 24, [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s]], [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s]], Yes, Yes, Yes, No, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims)
First verify the full flow feasibility conditions. Compare its value with a cut capacity and retain all equal-valued/minimizing cuts. The universal flow≤cut bound makes both optimal. The final No withholds integrality and algorithm claims, not the optimality certificate. The exact result, retaining the requested order, is (24, 24, [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s]], [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s]], Yes, Yes, Yes, No, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims).
Reviewed step-by-step derivation
Max-flow min-cut theorem · qf-mod46-mfmc-equal-value-optimality · candidate
82. An Equal-Valued Feasible Flow and Cut Certify Both Optima
Use complete directed vertices=[s, a0, a1, a2, a3, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t], [s, a2], [a2, t], [s, a3], [a3, t], [a0, s]], source=s, sink=t, aligned nonnegative exact capacities=[8, 8, 9, 9, 10, 10, 11, 11, 13], candidate flows=[8, 8, 9, 9, 10, 10, 11, 11, 0]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (flow_value, minimum_capacity, equal_value_cuts, minimum_cuts, equal_values, maximum_flow_certificate, minimum_cut_certificate, integrality_or_algorithm_claim, certificate).
Equal-value certification first requires a feasible flow. The final No withholds integrality or a particular algorithm proof, not maximum-flow/minimum-cut certification.
View answer and explanation
(38, 38, [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s], [a3, s], [a0, a3, s], [a1, a3, s], [a0, a1, a3, s], [a2, a3, s], [a0, a2, a3, s], [a1, a2, a3, s], [a0, a1, a2, a3, s]], [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s], [a3, s], [a0, a3, s], [a1, a3, s], [a0, a1, a3, s], [a2, a3, s], [a0, a2, a3, s], [a1, a2, a3, s], [a0, a1, a2, a3, s]], Yes, Yes, Yes, No, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims)
First verify the full flow feasibility conditions. Compare its value with a cut capacity and retain all equal-valued/minimizing cuts. The universal flow≤cut bound makes both optimal. The final No withholds integrality and algorithm claims, not the optimality certificate. The exact result, retaining the requested order, is (38, 38, [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s], [a3, s], [a0, a3, s], [a1, a3, s], [a0, a1, a3, s], [a2, a3, s], [a0, a2, a3, s], [a1, a2, a3, s], [a0, a1, a2, a3, s]], [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s], [a3, s], [a0, a3, s], [a1, a3, s], [a0, a1, a3, s], [a2, a3, s], [a0, a2, a3, s], [a1, a2, a3, s], [a0, a1, a2, a3, s]], Yes, Yes, Yes, No, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims).
Reviewed step-by-step derivation
Max-flow min-cut theorem · qf-mod46-mfmc-equal-value-optimality · candidate
83. An Equal-Valued Feasible Flow and Cut Certify Both Optima
Use complete directed vertices=[s, a0, a1, a2, a3, a4, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t], [s, a2], [a2, t], [s, a3], [a3, t], [s, a4], [a4, t]], source=s, sink=t, aligned nonnegative exact capacities=[9, 9, 10, 10, 11, 11, 12, 12, 13, 13], candidate flows=[9, 9, 10, 10, 11, 11, 12, 12, 13, 13]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (flow_value, minimum_capacity, equal_value_cuts, minimum_cuts, equal_values, maximum_flow_certificate, minimum_cut_certificate, integrality_or_algorithm_claim, certificate).
Equal-value certification first requires a feasible flow. The final No withholds integrality or a particular algorithm proof, not maximum-flow/minimum-cut certification.
View answer and explanation
(55, 55, [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s], [a3, s], [a0, a3, s], [a1, a3, s], [a0, a1, a3, s], [a2, a3, s], [a0, a2, a3, s], [a1, a2, a3, s], [a0, a1, a2, a3, s], [a4, s], [a0, a4, s], [a1, a4, s], [a0, a1, a4, s], [a2, a4, s], [a0, a2, a4, s], [a1, a2, a4, s], [a0, a1, a2, a4, s], [a3, a4, s], [a0, a3, a4, s], [a1, a3, a4, s], [a0, a1, a3, a4, s], [a2, a3, a4, s], [a0, a2, a3, a4, s], [a1, a2, a3, a4, s], [a0, a1, a2, a3, a4, s]], [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s], [a3, s], [a0, a3, s], [a1, a3, s], [a0, a1, a3, s], [a2, a3, s], [a0, a2, a3, s], [a1, a2, a3, s], [a0, a1, a2, a3, s], [a4, s], [a0, a4, s], [a1, a4, s], [a0, a1, a4, s], [a2, a4, s], [a0, a2, a4, s], [a1, a2, a4, s], [a0, a1, a2, a4, s], [a3, a4, s], [a0, a3, a4, s], [a1, a3, a4, s], [a0, a1, a3, a4, s], [a2, a3, a4, s], [a0, a2, a3, a4, s], [a1, a2, a3, a4, s], [a0, a1, a2, a3, a4, s]], Yes, Yes, Yes, No, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims)
First verify the full flow feasibility conditions. Compare its value with a cut capacity and retain all equal-valued/minimizing cuts. The universal flow≤cut bound makes both optimal. The final No withholds integrality and algorithm claims, not the optimality certificate. The exact result, retaining the requested order, is (55, 55, [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s], [a3, s], [a0, a3, s], [a1, a3, s], [a0, a1, a3, s], [a2, a3, s], [a0, a2, a3, s], [a1, a2, a3, s], [a0, a1, a2, a3, s], [a4, s], [a0, a4, s], [a1, a4, s], [a0, a1, a4, s], [a2, a4, s], [a0, a2, a4, s], [a1, a2, a4, s], [a0, a1, a2, a4, s], [a3, a4, s], [a0, a3, a4, s], [a1, a3, a4, s], [a0, a1, a3, a4, s], [a2, a3, a4, s], [a0, a2, a3, a4, s], [a1, a2, a3, a4, s], [a0, a1, a2, a3, a4, s]], [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s], [a3, s], [a0, a3, s], [a1, a3, s], [a0, a1, a3, s], [a2, a3, s], [a0, a2, a3, s], [a1, a2, a3, s], [a0, a1, a2, a3, s], [a4, s], [a0, a4, s], [a1, a4, s], [a0, a1, a4, s], [a2, a4, s], [a0, a2, a4, s], [a1, a2, a4, s], [a0, a1, a2, a4, s], [a3, a4, s], [a0, a3, a4, s], [a1, a3, a4, s], [a0, a1, a3, a4, s], [a2, a3, a4, s], [a0, a2, a3, a4, s], [a1, a2, a3, a4, s], [a0, a1, a2, a3, a4, s]], Yes, Yes, Yes, No, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims).
Reviewed step-by-step derivation
Max-flow min-cut theorem · qf-mod46-mfmc-equal-value-optimality · candidate
84. An Equal-Valued Feasible Flow and Cut Certify Both Optima
Use complete directed vertices=[s, a0, a1, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t]], source=s, sink=t, aligned nonnegative exact capacities=[10, 10, 11, 11], candidate flows=[10, 10, 11, 11]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (flow_value, minimum_capacity, equal_value_cuts, minimum_cuts, equal_values, maximum_flow_certificate, minimum_cut_certificate, integrality_or_algorithm_claim, certificate).
Equal-value certification first requires a feasible flow. The final No withholds integrality or a particular algorithm proof, not maximum-flow/minimum-cut certification.
View answer and explanation
(21, 21, [[s], [a0, s], [a1, s], [a0, a1, s]], [[s], [a0, s], [a1, s], [a0, a1, s]], Yes, Yes, Yes, No, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims)
First verify the full flow feasibility conditions. Compare its value with a cut capacity and retain all equal-valued/minimizing cuts. The universal flow≤cut bound makes both optimal. The final No withholds integrality and algorithm claims, not the optimality certificate. The exact result, retaining the requested order, is (21, 21, [[s], [a0, s], [a1, s], [a0, a1, s]], [[s], [a0, s], [a1, s], [a0, a1, s]], Yes, Yes, Yes, No, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims).
Reviewed step-by-step derivation
Max-flow min-cut theorem · qf-mod46-mfmc-equal-value-optimality · candidate
85. An Equal-Valued Feasible Flow and Cut Certify Both Optima
Use complete directed vertices=[s, a0, a1, a2, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t], [s, a2], [a2, t], [a0, s]], source=s, sink=t, aligned nonnegative exact capacities=[11, 11, 12, 12, 13, 13, 16], candidate flows=[11, 11, 12, 12, 13, 13, 0]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (flow_value, minimum_capacity, equal_value_cuts, minimum_cuts, equal_values, maximum_flow_certificate, minimum_cut_certificate, integrality_or_algorithm_claim, certificate).
Equal-value certification first requires a feasible flow. The final No withholds integrality or a particular algorithm proof, not maximum-flow/minimum-cut certification.
View answer and explanation
(36, 36, [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s]], [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s]], Yes, Yes, Yes, No, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims)
First verify the full flow feasibility conditions. Compare its value with a cut capacity and retain all equal-valued/minimizing cuts. The universal flow≤cut bound makes both optimal. The final No withholds integrality and algorithm claims, not the optimality certificate. The exact result, retaining the requested order, is (36, 36, [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s]], [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s]], Yes, Yes, Yes, No, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims).
Reviewed step-by-step derivation
Max-flow min-cut theorem · qf-mod46-mfmc-equal-value-optimality · candidate
86. An Equal-Valued Feasible Flow and Cut Certify Both Optima
Use complete directed vertices=[s, a0, a1, a2, a3, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t], [s, a2], [a2, t], [s, a3], [a3, t]], source=s, sink=t, aligned nonnegative exact capacities=[12, 12, 13, 13, 14, 14, 15, 15], candidate flows=[12, 12, 13, 13, 14, 14, 15, 15]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (flow_value, minimum_capacity, equal_value_cuts, minimum_cuts, equal_values, maximum_flow_certificate, minimum_cut_certificate, integrality_or_algorithm_claim, certificate).
Equal-value certification first requires a feasible flow. The final No withholds integrality or a particular algorithm proof, not maximum-flow/minimum-cut certification.
View answer and explanation
(54, 54, [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s], [a3, s], [a0, a3, s], [a1, a3, s], [a0, a1, a3, s], [a2, a3, s], [a0, a2, a3, s], [a1, a2, a3, s], [a0, a1, a2, a3, s]], [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s], [a3, s], [a0, a3, s], [a1, a3, s], [a0, a1, a3, s], [a2, a3, s], [a0, a2, a3, s], [a1, a2, a3, s], [a0, a1, a2, a3, s]], Yes, Yes, Yes, No, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims)
First verify the full flow feasibility conditions. Compare its value with a cut capacity and retain all equal-valued/minimizing cuts. The universal flow≤cut bound makes both optimal. The final No withholds integrality and algorithm claims, not the optimality certificate. The exact result, retaining the requested order, is (54, 54, [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s], [a3, s], [a0, a3, s], [a1, a3, s], [a0, a1, a3, s], [a2, a3, s], [a0, a2, a3, s], [a1, a2, a3, s], [a0, a1, a2, a3, s]], [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s], [a3, s], [a0, a3, s], [a1, a3, s], [a0, a1, a3, s], [a2, a3, s], [a0, a2, a3, s], [a1, a2, a3, s], [a0, a1, a2, a3, s]], Yes, Yes, Yes, No, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims).
Reviewed step-by-step derivation
Max-flow min-cut theorem · qf-mod46-mfmc-equal-value-optimality · candidate
87. An Equal-Valued Feasible Flow and Cut Certify Both Optima
Use complete directed vertices=[s, a0, a1, a2, a3, a4, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t], [s, a2], [a2, t], [s, a3], [a3, t], [s, a4], [a4, t]], source=s, sink=t, aligned nonnegative exact capacities=[13, 13, 14, 14, 15, 15, 16, 16, 17, 17], candidate flows=[13, 13, 14, 14, 15, 15, 16, 16, 17, 17]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (flow_value, minimum_capacity, equal_value_cuts, minimum_cuts, equal_values, maximum_flow_certificate, minimum_cut_certificate, integrality_or_algorithm_claim, certificate).
Equal-value certification first requires a feasible flow. The final No withholds integrality or a particular algorithm proof, not maximum-flow/minimum-cut certification.
View answer and explanation
(75, 75, [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s], [a3, s], [a0, a3, s], [a1, a3, s], [a0, a1, a3, s], [a2, a3, s], [a0, a2, a3, s], [a1, a2, a3, s], [a0, a1, a2, a3, s], [a4, s], [a0, a4, s], [a1, a4, s], [a0, a1, a4, s], [a2, a4, s], [a0, a2, a4, s], [a1, a2, a4, s], [a0, a1, a2, a4, s], [a3, a4, s], [a0, a3, a4, s], [a1, a3, a4, s], [a0, a1, a3, a4, s], [a2, a3, a4, s], [a0, a2, a3, a4, s], [a1, a2, a3, a4, s], [a0, a1, a2, a3, a4, s]], [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s], [a3, s], [a0, a3, s], [a1, a3, s], [a0, a1, a3, s], [a2, a3, s], [a0, a2, a3, s], [a1, a2, a3, s], [a0, a1, a2, a3, s], [a4, s], [a0, a4, s], [a1, a4, s], [a0, a1, a4, s], [a2, a4, s], [a0, a2, a4, s], [a1, a2, a4, s], [a0, a1, a2, a4, s], [a3, a4, s], [a0, a3, a4, s], [a1, a3, a4, s], [a0, a1, a3, a4, s], [a2, a3, a4, s], [a0, a2, a3, a4, s], [a1, a2, a3, a4, s], [a0, a1, a2, a3, a4, s]], Yes, Yes, Yes, No, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims)
First verify the full flow feasibility conditions. Compare its value with a cut capacity and retain all equal-valued/minimizing cuts. The universal flow≤cut bound makes both optimal. The final No withholds integrality and algorithm claims, not the optimality certificate. The exact result, retaining the requested order, is (75, 75, [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s], [a3, s], [a0, a3, s], [a1, a3, s], [a0, a1, a3, s], [a2, a3, s], [a0, a2, a3, s], [a1, a2, a3, s], [a0, a1, a2, a3, s], [a4, s], [a0, a4, s], [a1, a4, s], [a0, a1, a4, s], [a2, a4, s], [a0, a2, a4, s], [a1, a2, a4, s], [a0, a1, a2, a4, s], [a3, a4, s], [a0, a3, a4, s], [a1, a3, a4, s], [a0, a1, a3, a4, s], [a2, a3, a4, s], [a0, a2, a3, a4, s], [a1, a2, a3, a4, s], [a0, a1, a2, a3, a4, s]], [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s], [a3, s], [a0, a3, s], [a1, a3, s], [a0, a1, a3, s], [a2, a3, s], [a0, a2, a3, s], [a1, a2, a3, s], [a0, a1, a2, a3, s], [a4, s], [a0, a4, s], [a1, a4, s], [a0, a1, a4, s], [a2, a4, s], [a0, a2, a4, s], [a1, a2, a4, s], [a0, a1, a2, a4, s], [a3, a4, s], [a0, a3, a4, s], [a1, a3, a4, s], [a0, a1, a3, a4, s], [a2, a3, a4, s], [a0, a2, a3, a4, s], [a1, a2, a3, a4, s], [a0, a1, a2, a3, a4, s]], Yes, Yes, Yes, No, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims).
Reviewed step-by-step derivation
Max-flow min-cut theorem · qf-mod46-mfmc-equal-value-optimality · candidate
88. An Equal-Valued Feasible Flow and Cut Certify Both Optima
Use complete directed vertices=[s, a0, a1, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t], [a0, s]], source=s, sink=t, aligned nonnegative exact capacities=[14, 14, 15, 15, 19], candidate flows=[14, 14, 15, 15, 0]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (flow_value, minimum_capacity, equal_value_cuts, minimum_cuts, equal_values, maximum_flow_certificate, minimum_cut_certificate, integrality_or_algorithm_claim, certificate).
Equal-value certification first requires a feasible flow. The final No withholds integrality or a particular algorithm proof, not maximum-flow/minimum-cut certification.
View answer and explanation
(29, 29, [[s], [a0, s], [a1, s], [a0, a1, s]], [[s], [a0, s], [a1, s], [a0, a1, s]], Yes, Yes, Yes, No, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims)
First verify the full flow feasibility conditions. Compare its value with a cut capacity and retain all equal-valued/minimizing cuts. The universal flow≤cut bound makes both optimal. The final No withholds integrality and algorithm claims, not the optimality certificate. The exact result, retaining the requested order, is (29, 29, [[s], [a0, s], [a1, s], [a0, a1, s]], [[s], [a0, s], [a1, s], [a0, a1, s]], Yes, Yes, Yes, No, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims).
Reviewed step-by-step derivation
Max-flow min-cut theorem · qf-mod46-mfmc-equal-value-optimality · candidate
89. An Equal-Valued Feasible Flow and Cut Certify Both Optima
Use complete directed vertices=[s, a0, a1, a2, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t], [s, a2], [a2, t]], source=s, sink=t, aligned nonnegative exact capacities=[15, 15, 16, 16, 17, 17], candidate flows=[15, 15, 16, 16, 17, 17]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (flow_value, minimum_capacity, equal_value_cuts, minimum_cuts, equal_values, maximum_flow_certificate, minimum_cut_certificate, integrality_or_algorithm_claim, certificate).
Equal-value certification first requires a feasible flow. The final No withholds integrality or a particular algorithm proof, not maximum-flow/minimum-cut certification.
View answer and explanation
(48, 48, [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s]], [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s]], Yes, Yes, Yes, No, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims)
First verify the full flow feasibility conditions. Compare its value with a cut capacity and retain all equal-valued/minimizing cuts. The universal flow≤cut bound makes both optimal. The final No withholds integrality and algorithm claims, not the optimality certificate. The exact result, retaining the requested order, is (48, 48, [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s]], [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s]], Yes, Yes, Yes, No, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims).
Reviewed step-by-step derivation
Max-flow min-cut theorem · qf-mod46-mfmc-equal-value-optimality · candidate
90. An Equal-Valued Feasible Flow and Cut Certify Both Optima
Use complete directed vertices=[s, a0, a1, a2, a3, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t], [s, a2], [a2, t], [s, a3], [a3, t]], source=s, sink=t, aligned nonnegative exact capacities=[16, 16, 17, 17, 18, 18, 19, 19], candidate flows=[16, 16, 17, 17, 18, 18, 19, 19]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (flow_value, minimum_capacity, equal_value_cuts, minimum_cuts, equal_values, maximum_flow_certificate, minimum_cut_certificate, integrality_or_algorithm_claim, certificate).
Equal-value certification first requires a feasible flow. The final No withholds integrality or a particular algorithm proof, not maximum-flow/minimum-cut certification.
View answer and explanation
(70, 70, [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s], [a3, s], [a0, a3, s], [a1, a3, s], [a0, a1, a3, s], [a2, a3, s], [a0, a2, a3, s], [a1, a2, a3, s], [a0, a1, a2, a3, s]], [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s], [a3, s], [a0, a3, s], [a1, a3, s], [a0, a1, a3, s], [a2, a3, s], [a0, a2, a3, s], [a1, a2, a3, s], [a0, a1, a2, a3, s]], Yes, Yes, Yes, No, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims)
First verify the full flow feasibility conditions. Compare its value with a cut capacity and retain all equal-valued/minimizing cuts. The universal flow≤cut bound makes both optimal. The final No withholds integrality and algorithm claims, not the optimality certificate. The exact result, retaining the requested order, is (70, 70, [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s], [a3, s], [a0, a3, s], [a1, a3, s], [a0, a1, a3, s], [a2, a3, s], [a0, a2, a3, s], [a1, a2, a3, s], [a0, a1, a2, a3, s]], [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s], [a3, s], [a0, a3, s], [a1, a3, s], [a0, a1, a3, s], [a2, a3, s], [a0, a2, a3, s], [a1, a2, a3, s], [a0, a1, a2, a3, s]], Yes, Yes, Yes, No, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims).
Reviewed step-by-step derivation
Max-flow min-cut theorem · qf-mod46-mfmc-equal-value-optimality · candidate
91. An Equal-Valued Feasible Flow and Cut Certify Both Optima
Use complete directed vertices=[s, a0, a1, a2, a3, a4, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t], [s, a2], [a2, t], [s, a3], [a3, t], [s, a4], [a4, t], [a0, s]], source=s, sink=t, aligned nonnegative exact capacities=[17, 17, 18, 18, 19, 19, 20, 20, 21, 21, 22], candidate flows=[17, 17, 18, 18, 19, 19, 20, 20, 21, 21, 0]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (flow_value, minimum_capacity, equal_value_cuts, minimum_cuts, equal_values, maximum_flow_certificate, minimum_cut_certificate, integrality_or_algorithm_claim, certificate).
Equal-value certification first requires a feasible flow. The final No withholds integrality or a particular algorithm proof, not maximum-flow/minimum-cut certification.
View answer and explanation
(95, 95, [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s], [a3, s], [a0, a3, s], [a1, a3, s], [a0, a1, a3, s], [a2, a3, s], [a0, a2, a3, s], [a1, a2, a3, s], [a0, a1, a2, a3, s], [a4, s], [a0, a4, s], [a1, a4, s], [a0, a1, a4, s], [a2, a4, s], [a0, a2, a4, s], [a1, a2, a4, s], [a0, a1, a2, a4, s], [a3, a4, s], [a0, a3, a4, s], [a1, a3, a4, s], [a0, a1, a3, a4, s], [a2, a3, a4, s], [a0, a2, a3, a4, s], [a1, a2, a3, a4, s], [a0, a1, a2, a3, a4, s]], [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s], [a3, s], [a0, a3, s], [a1, a3, s], [a0, a1, a3, s], [a2, a3, s], [a0, a2, a3, s], [a1, a2, a3, s], [a0, a1, a2, a3, s], [a4, s], [a0, a4, s], [a1, a4, s], [a0, a1, a4, s], [a2, a4, s], [a0, a2, a4, s], [a1, a2, a4, s], [a0, a1, a2, a4, s], [a3, a4, s], [a0, a3, a4, s], [a1, a3, a4, s], [a0, a1, a3, a4, s], [a2, a3, a4, s], [a0, a2, a3, a4, s], [a1, a2, a3, a4, s], [a0, a1, a2, a3, a4, s]], Yes, Yes, Yes, No, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims)
First verify the full flow feasibility conditions. Compare its value with a cut capacity and retain all equal-valued/minimizing cuts. The universal flow≤cut bound makes both optimal. The final No withholds integrality and algorithm claims, not the optimality certificate. The exact result, retaining the requested order, is (95, 95, [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s], [a3, s], [a0, a3, s], [a1, a3, s], [a0, a1, a3, s], [a2, a3, s], [a0, a2, a3, s], [a1, a2, a3, s], [a0, a1, a2, a3, s], [a4, s], [a0, a4, s], [a1, a4, s], [a0, a1, a4, s], [a2, a4, s], [a0, a2, a4, s], [a1, a2, a4, s], [a0, a1, a2, a4, s], [a3, a4, s], [a0, a3, a4, s], [a1, a3, a4, s], [a0, a1, a3, a4, s], [a2, a3, a4, s], [a0, a2, a3, a4, s], [a1, a2, a3, a4, s], [a0, a1, a2, a3, a4, s]], [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s], [a3, s], [a0, a3, s], [a1, a3, s], [a0, a1, a3, s], [a2, a3, s], [a0, a2, a3, s], [a1, a2, a3, s], [a0, a1, a2, a3, s], [a4, s], [a0, a4, s], [a1, a4, s], [a0, a1, a4, s], [a2, a4, s], [a0, a2, a4, s], [a1, a2, a4, s], [a0, a1, a2, a4, s], [a3, a4, s], [a0, a3, a4, s], [a1, a3, a4, s], [a0, a1, a3, a4, s], [a2, a3, a4, s], [a0, a2, a3, a4, s], [a1, a2, a3, a4, s], [a0, a1, a2, a3, a4, s]], Yes, Yes, Yes, No, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims).
Reviewed step-by-step derivation
Max-flow min-cut theorem · qf-mod46-mfmc-equal-value-optimality · candidate
92. An Equal-Valued Feasible Flow and Cut Certify Both Optima
Use complete directed vertices=[s, a0, a1, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t]], source=s, sink=t, aligned nonnegative exact capacities=[18, 18, 19, 19], candidate flows=[18, 18, 19, 19]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (flow_value, minimum_capacity, equal_value_cuts, minimum_cuts, equal_values, maximum_flow_certificate, minimum_cut_certificate, integrality_or_algorithm_claim, certificate).
Equal-value certification first requires a feasible flow. The final No withholds integrality or a particular algorithm proof, not maximum-flow/minimum-cut certification.
View answer and explanation
(37, 37, [[s], [a0, s], [a1, s], [a0, a1, s]], [[s], [a0, s], [a1, s], [a0, a1, s]], Yes, Yes, Yes, No, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims)
First verify the full flow feasibility conditions. Compare its value with a cut capacity and retain all equal-valued/minimizing cuts. The universal flow≤cut bound makes both optimal. The final No withholds integrality and algorithm claims, not the optimality certificate. The exact result, retaining the requested order, is (37, 37, [[s], [a0, s], [a1, s], [a0, a1, s]], [[s], [a0, s], [a1, s], [a0, a1, s]], Yes, Yes, Yes, No, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims).
Reviewed step-by-step derivation
Max-flow min-cut theorem · qf-mod46-mfmc-equal-value-optimality · candidate
93. An Equal-Valued Feasible Flow and Cut Certify Both Optima
Use complete directed vertices=[s, a0, a1, a2, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t], [s, a2], [a2, t]], source=s, sink=t, aligned nonnegative exact capacities=[19, 19, 20, 20, 21, 21], candidate flows=[19, 19, 20, 20, 21, 21]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (flow_value, minimum_capacity, equal_value_cuts, minimum_cuts, equal_values, maximum_flow_certificate, minimum_cut_certificate, integrality_or_algorithm_claim, certificate).
Equal-value certification first requires a feasible flow. The final No withholds integrality or a particular algorithm proof, not maximum-flow/minimum-cut certification.
View answer and explanation
(60, 60, [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s]], [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s]], Yes, Yes, Yes, No, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims)
First verify the full flow feasibility conditions. Compare its value with a cut capacity and retain all equal-valued/minimizing cuts. The universal flow≤cut bound makes both optimal. The final No withholds integrality and algorithm claims, not the optimality certificate. The exact result, retaining the requested order, is (60, 60, [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s]], [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s]], Yes, Yes, Yes, No, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims).
Reviewed step-by-step derivation
Max-flow min-cut theorem · qf-mod46-mfmc-equal-value-optimality · candidate
94. An Equal-Valued Feasible Flow and Cut Certify Both Optima
Use complete directed vertices=[s, a0, a1, a2, a3, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t], [s, a2], [a2, t], [s, a3], [a3, t], [a0, s]], source=s, sink=t, aligned nonnegative exact capacities=[20, 20, 21, 21, 22, 22, 23, 23, 25], candidate flows=[20, 20, 21, 21, 22, 22, 23, 23, 0]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (flow_value, minimum_capacity, equal_value_cuts, minimum_cuts, equal_values, maximum_flow_certificate, minimum_cut_certificate, integrality_or_algorithm_claim, certificate).
Equal-value certification first requires a feasible flow. The final No withholds integrality or a particular algorithm proof, not maximum-flow/minimum-cut certification.
View answer and explanation
(86, 86, [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s], [a3, s], [a0, a3, s], [a1, a3, s], [a0, a1, a3, s], [a2, a3, s], [a0, a2, a3, s], [a1, a2, a3, s], [a0, a1, a2, a3, s]], [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s], [a3, s], [a0, a3, s], [a1, a3, s], [a0, a1, a3, s], [a2, a3, s], [a0, a2, a3, s], [a1, a2, a3, s], [a0, a1, a2, a3, s]], Yes, Yes, Yes, No, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims)
First verify the full flow feasibility conditions. Compare its value with a cut capacity and retain all equal-valued/minimizing cuts. The universal flow≤cut bound makes both optimal. The final No withholds integrality and algorithm claims, not the optimality certificate. The exact result, retaining the requested order, is (86, 86, [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s], [a3, s], [a0, a3, s], [a1, a3, s], [a0, a1, a3, s], [a2, a3, s], [a0, a2, a3, s], [a1, a2, a3, s], [a0, a1, a2, a3, s]], [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s], [a3, s], [a0, a3, s], [a1, a3, s], [a0, a1, a3, s], [a2, a3, s], [a0, a2, a3, s], [a1, a2, a3, s], [a0, a1, a2, a3, s]], Yes, Yes, Yes, No, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims).
Reviewed step-by-step derivation
Max-flow min-cut theorem · qf-mod46-mfmc-equal-value-optimality · candidate
95. An Equal-Valued Feasible Flow and Cut Certify Both Optima
Use complete directed vertices=[s, a0, a1, a2, a3, a4, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t], [s, a2], [a2, t], [s, a3], [a3, t], [s, a4], [a4, t]], source=s, sink=t, aligned nonnegative exact capacities=[21, 21, 22, 22, 23, 23, 24, 24, 25, 25], candidate flows=[21, 21, 22, 22, 23, 23, 24, 24, 25, 25]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (flow_value, minimum_capacity, equal_value_cuts, minimum_cuts, equal_values, maximum_flow_certificate, minimum_cut_certificate, integrality_or_algorithm_claim, certificate).
Equal-value certification first requires a feasible flow. The final No withholds integrality or a particular algorithm proof, not maximum-flow/minimum-cut certification.
View answer and explanation
(115, 115, [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s], [a3, s], [a0, a3, s], [a1, a3, s], [a0, a1, a3, s], [a2, a3, s], [a0, a2, a3, s], [a1, a2, a3, s], [a0, a1, a2, a3, s], [a4, s], [a0, a4, s], [a1, a4, s], [a0, a1, a4, s], [a2, a4, s], [a0, a2, a4, s], [a1, a2, a4, s], [a0, a1, a2, a4, s], [a3, a4, s], [a0, a3, a4, s], [a1, a3, a4, s], [a0, a1, a3, a4, s], [a2, a3, a4, s], [a0, a2, a3, a4, s], [a1, a2, a3, a4, s], [a0, a1, a2, a3, a4, s]], [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s], [a3, s], [a0, a3, s], [a1, a3, s], [a0, a1, a3, s], [a2, a3, s], [a0, a2, a3, s], [a1, a2, a3, s], [a0, a1, a2, a3, s], [a4, s], [a0, a4, s], [a1, a4, s], [a0, a1, a4, s], [a2, a4, s], [a0, a2, a4, s], [a1, a2, a4, s], [a0, a1, a2, a4, s], [a3, a4, s], [a0, a3, a4, s], [a1, a3, a4, s], [a0, a1, a3, a4, s], [a2, a3, a4, s], [a0, a2, a3, a4, s], [a1, a2, a3, a4, s], [a0, a1, a2, a3, a4, s]], Yes, Yes, Yes, No, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims)
First verify the full flow feasibility conditions. Compare its value with a cut capacity and retain all equal-valued/minimizing cuts. The universal flow≤cut bound makes both optimal. The final No withholds integrality and algorithm claims, not the optimality certificate. The exact result, retaining the requested order, is (115, 115, [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s], [a3, s], [a0, a3, s], [a1, a3, s], [a0, a1, a3, s], [a2, a3, s], [a0, a2, a3, s], [a1, a2, a3, s], [a0, a1, a2, a3, s], [a4, s], [a0, a4, s], [a1, a4, s], [a0, a1, a4, s], [a2, a4, s], [a0, a2, a4, s], [a1, a2, a4, s], [a0, a1, a2, a4, s], [a3, a4, s], [a0, a3, a4, s], [a1, a3, a4, s], [a0, a1, a3, a4, s], [a2, a3, a4, s], [a0, a2, a3, a4, s], [a1, a2, a3, a4, s], [a0, a1, a2, a3, a4, s]], [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s], [a3, s], [a0, a3, s], [a1, a3, s], [a0, a1, a3, s], [a2, a3, s], [a0, a2, a3, s], [a1, a2, a3, s], [a0, a1, a2, a3, s], [a4, s], [a0, a4, s], [a1, a4, s], [a0, a1, a4, s], [a2, a4, s], [a0, a2, a4, s], [a1, a2, a4, s], [a0, a1, a2, a4, s], [a3, a4, s], [a0, a3, a4, s], [a1, a3, a4, s], [a0, a1, a3, a4, s], [a2, a3, a4, s], [a0, a2, a3, a4, s], [a1, a2, a3, a4, s], [a0, a1, a2, a3, a4, s]], Yes, Yes, Yes, No, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims).
Reviewed step-by-step derivation
Max-flow min-cut theorem · qf-mod46-mfmc-equal-value-optimality · candidate
96. An Equal-Valued Feasible Flow and Cut Certify Both Optima
Use complete directed vertices=[s, a0, a1, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t]], source=s, sink=t, aligned nonnegative exact capacities=[22, 22, 23, 23], candidate flows=[22, 22, 23, 23]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (flow_value, minimum_capacity, equal_value_cuts, minimum_cuts, equal_values, maximum_flow_certificate, minimum_cut_certificate, integrality_or_algorithm_claim, certificate).
Equal-value certification first requires a feasible flow. The final No withholds integrality or a particular algorithm proof, not maximum-flow/minimum-cut certification.
View answer and explanation
(45, 45, [[s], [a0, s], [a1, s], [a0, a1, s]], [[s], [a0, s], [a1, s], [a0, a1, s]], Yes, Yes, Yes, No, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims)
First verify the full flow feasibility conditions. Compare its value with a cut capacity and retain all equal-valued/minimizing cuts. The universal flow≤cut bound makes both optimal. The final No withholds integrality and algorithm claims, not the optimality certificate. The exact result, retaining the requested order, is (45, 45, [[s], [a0, s], [a1, s], [a0, a1, s]], [[s], [a0, s], [a1, s], [a0, a1, s]], Yes, Yes, Yes, No, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims).
Reviewed step-by-step derivation
Max-flow min-cut theorem · qf-mod46-mfmc-equal-value-optimality · candidate
97. An Equal-Valued Feasible Flow and Cut Certify Both Optima
Use complete directed vertices=[s, a0, a1, a2, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t], [s, a2], [a2, t], [a0, s]], source=s, sink=t, aligned nonnegative exact capacities=[23, 23, 24, 24, 25, 25, 28], candidate flows=[23, 23, 24, 24, 25, 25, 0]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (flow_value, minimum_capacity, equal_value_cuts, minimum_cuts, equal_values, maximum_flow_certificate, minimum_cut_certificate, integrality_or_algorithm_claim, certificate).
Equal-value certification first requires a feasible flow. The final No withholds integrality or a particular algorithm proof, not maximum-flow/minimum-cut certification.
View answer and explanation
(72, 72, [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s]], [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s]], Yes, Yes, Yes, No, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims)
First verify the full flow feasibility conditions. Compare its value with a cut capacity and retain all equal-valued/minimizing cuts. The universal flow≤cut bound makes both optimal. The final No withholds integrality and algorithm claims, not the optimality certificate. The exact result, retaining the requested order, is (72, 72, [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s]], [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s]], Yes, Yes, Yes, No, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims).
Reviewed step-by-step derivation
Max-flow min-cut theorem · qf-mod46-mfmc-equal-value-optimality · candidate
98. An Equal-Valued Feasible Flow and Cut Certify Both Optima
Use complete directed vertices=[s, a0, a1, a2, a3, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t], [s, a2], [a2, t], [s, a3], [a3, t]], source=s, sink=t, aligned nonnegative exact capacities=[24, 24, 25, 25, 26, 26, 27, 27], candidate flows=[24, 24, 25, 25, 26, 26, 27, 27]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (flow_value, minimum_capacity, equal_value_cuts, minimum_cuts, equal_values, maximum_flow_certificate, minimum_cut_certificate, integrality_or_algorithm_claim, certificate).
Equal-value certification first requires a feasible flow. The final No withholds integrality or a particular algorithm proof, not maximum-flow/minimum-cut certification.
View answer and explanation
(102, 102, [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s], [a3, s], [a0, a3, s], [a1, a3, s], [a0, a1, a3, s], [a2, a3, s], [a0, a2, a3, s], [a1, a2, a3, s], [a0, a1, a2, a3, s]], [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s], [a3, s], [a0, a3, s], [a1, a3, s], [a0, a1, a3, s], [a2, a3, s], [a0, a2, a3, s], [a1, a2, a3, s], [a0, a1, a2, a3, s]], Yes, Yes, Yes, No, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims)
First verify the full flow feasibility conditions. Compare its value with a cut capacity and retain all equal-valued/minimizing cuts. The universal flow≤cut bound makes both optimal. The final No withholds integrality and algorithm claims, not the optimality certificate. The exact result, retaining the requested order, is (102, 102, [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s], [a3, s], [a0, a3, s], [a1, a3, s], [a0, a1, a3, s], [a2, a3, s], [a0, a2, a3, s], [a1, a2, a3, s], [a0, a1, a2, a3, s]], [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s], [a3, s], [a0, a3, s], [a1, a3, s], [a0, a1, a3, s], [a2, a3, s], [a0, a2, a3, s], [a1, a2, a3, s], [a0, a1, a2, a3, s]], Yes, Yes, Yes, No, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims).
Reviewed step-by-step derivation
Max-flow min-cut theorem · qf-mod46-mfmc-equal-value-optimality · candidate
99. An Equal-Valued Feasible Flow and Cut Certify Both Optima
Use complete directed vertices=[s, a0, a1, a2, a3, a4, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t], [s, a2], [a2, t], [s, a3], [a3, t], [s, a4], [a4, t]], source=s, sink=t, aligned nonnegative exact capacities=[25, 25, 26, 26, 27, 27, 28, 28, 29, 29], candidate flows=[25, 25, 26, 26, 27, 27, 28, 28, 29, 29]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (flow_value, minimum_capacity, equal_value_cuts, minimum_cuts, equal_values, maximum_flow_certificate, minimum_cut_certificate, integrality_or_algorithm_claim, certificate).
Equal-value certification first requires a feasible flow. The final No withholds integrality or a particular algorithm proof, not maximum-flow/minimum-cut certification.
View answer and explanation
(135, 135, [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s], [a3, s], [a0, a3, s], [a1, a3, s], [a0, a1, a3, s], [a2, a3, s], [a0, a2, a3, s], [a1, a2, a3, s], [a0, a1, a2, a3, s], [a4, s], [a0, a4, s], [a1, a4, s], [a0, a1, a4, s], [a2, a4, s], [a0, a2, a4, s], [a1, a2, a4, s], [a0, a1, a2, a4, s], [a3, a4, s], [a0, a3, a4, s], [a1, a3, a4, s], [a0, a1, a3, a4, s], [a2, a3, a4, s], [a0, a2, a3, a4, s], [a1, a2, a3, a4, s], [a0, a1, a2, a3, a4, s]], [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s], [a3, s], [a0, a3, s], [a1, a3, s], [a0, a1, a3, s], [a2, a3, s], [a0, a2, a3, s], [a1, a2, a3, s], [a0, a1, a2, a3, s], [a4, s], [a0, a4, s], [a1, a4, s], [a0, a1, a4, s], [a2, a4, s], [a0, a2, a4, s], [a1, a2, a4, s], [a0, a1, a2, a4, s], [a3, a4, s], [a0, a3, a4, s], [a1, a3, a4, s], [a0, a1, a3, a4, s], [a2, a3, a4, s], [a0, a2, a3, a4, s], [a1, a2, a3, a4, s], [a0, a1, a2, a3, a4, s]], Yes, Yes, Yes, No, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims)
First verify the full flow feasibility conditions. Compare its value with a cut capacity and retain all equal-valued/minimizing cuts. The universal flow≤cut bound makes both optimal. The final No withholds integrality and algorithm claims, not the optimality certificate. The exact result, retaining the requested order, is (135, 135, [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s], [a3, s], [a0, a3, s], [a1, a3, s], [a0, a1, a3, s], [a2, a3, s], [a0, a2, a3, s], [a1, a2, a3, s], [a0, a1, a2, a3, s], [a4, s], [a0, a4, s], [a1, a4, s], [a0, a1, a4, s], [a2, a4, s], [a0, a2, a4, s], [a1, a2, a4, s], [a0, a1, a2, a4, s], [a3, a4, s], [a0, a3, a4, s], [a1, a3, a4, s], [a0, a1, a3, a4, s], [a2, a3, a4, s], [a0, a2, a3, a4, s], [a1, a2, a3, a4, s], [a0, a1, a2, a3, a4, s]], [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s], [a3, s], [a0, a3, s], [a1, a3, s], [a0, a1, a3, s], [a2, a3, s], [a0, a2, a3, s], [a1, a2, a3, s], [a0, a1, a2, a3, s], [a4, s], [a0, a4, s], [a1, a4, s], [a0, a1, a4, s], [a2, a4, s], [a0, a2, a4, s], [a1, a2, a4, s], [a0, a1, a2, a4, s], [a3, a4, s], [a0, a3, a4, s], [a1, a3, a4, s], [a0, a1, a3, a4, s], [a2, a3, a4, s], [a0, a2, a3, a4, s], [a1, a2, a3, a4, s], [a0, a1, a2, a3, a4, s]], Yes, Yes, Yes, No, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims).
Reviewed step-by-step derivation
Max-flow min-cut theorem · qf-mod46-mfmc-equal-value-optimality · candidate
100. An Equal-Valued Feasible Flow and Cut Certify Both Optima
Use complete directed vertices=[s, a0, a1, t], ordered edges=[[s, a0], [a0, t], [s, a1], [a1, t], [a0, s]], source=s, sink=t, aligned nonnegative exact capacities=[26, 26, 27, 27, 31], candidate flows=[26, 26, 27, 27, 0]. A cut contains the source and excludes the sink; its capacity sums only edges leaving the cut, not incoming edges. Give (flow_value, minimum_capacity, equal_value_cuts, minimum_cuts, equal_values, maximum_flow_certificate, minimum_cut_certificate, integrality_or_algorithm_claim, certificate).
Equal-value certification first requires a feasible flow. The final No withholds integrality or a particular algorithm proof, not maximum-flow/minimum-cut certification.
View answer and explanation
(53, 53, [[s], [a0, s], [a1, s], [a0, a1, s]], [[s], [a0, s], [a1, s], [a0, a1, s]], Yes, Yes, Yes, No, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims)
First verify the full flow feasibility conditions. Compare its value with a cut capacity and retain all equal-valued/minimizing cuts. The universal flow≤cut bound makes both optimal. The final No withholds integrality and algorithm claims, not the optimality certificate. The exact result, retaining the requested order, is (53, 53, [[s], [a0, s], [a1, s], [a0, a1, s]], [[s], [a0, s], [a1, s], [a0, a1, s]], Yes, Yes, Yes, No, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims).
Reviewed step-by-step derivation
Max-flow min-cut theorem · qf-mod46-mfmc-equal-value-optimality · candidate
Canonical question JSONL · English JSONL · Validation report