Question 25978: The Minimum Cut Is Selected from the Complete Cut Family

Original question and givens

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).

• G001 — 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).

Definition and scope

Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate.

• mko-max-flow-min-cut-theorem

Step-by-step derivation

source roles = [mfmc-minimum-cut-audit]

derived = [[cut_rows, minimum_capacity, minimum_cuts, certificate]]

source roles = [[s, a0, a1, a2, a3, t], [[s, a0], [a0, t], [s, a1], [a1, t], [s, a2], [a2, t], [s, a3], [a3, t]], [4, 4, 5, 5, 6, 6, 7, 7], null, s, t]

derived = [[a0, a1, a2, a3, s, t], [[a0, t], [a1, t], [a2, t], [a3, t], [s, a0], [s, a1], [s, a2], [s, a3]], [4, 5, 6, 7, 4, 5, 6, 7], null]

MFMC cut mask0 OUTWARD capacity := sum(4, 5, 6, 7)

MFMC cut mask0 OUTWARD capacity = 22

source roles = [0, [a0, a1, a2, a3], [s]]

derived = [[4, 5, 6, 7], [], [[[s, a0], 4], [[s, a1], 5], [[s, a2], 6], [[s, a3], 7]], 22]

MFMC cut mask1 OUTWARD capacity := sum(4, 5, 6, 7)

MFMC cut mask1 OUTWARD capacity = 22

source roles = [1, [a0, a1, a2, a3], [a0, s]]

derived = [[0, 5, 6, 7], [], [[[a0, t], 4], [[s, a1], 5], [[s, a2], 6], [[s, a3], 7]], 22]

MFMC cut mask2 OUTWARD capacity := sum(5, 4, 6, 7)

MFMC cut mask2 OUTWARD capacity = 22

source roles = [2, [a0, a1, a2, a3], [a1, s]]

derived = [[1, 4, 6, 7], [], [[[a1, t], 5], [[s, a0], 4], [[s, a2], 6], [[s, a3], 7]], 22]

MFMC cut mask3 OUTWARD capacity := sum(4, 5, 6, 7)

MFMC cut mask3 OUTWARD capacity = 22

source roles = [3, [a0, a1, a2, a3], [a0, a1, s]]

derived = [[0, 1, 6, 7], [], [[[a0, t], 4], [[a1, t], 5], [[s, a2], 6], [[s, a3], 7]], 22]

MFMC cut mask4 OUTWARD capacity := sum(6, 4, 5, 7)

MFMC cut mask4 OUTWARD capacity = 22

source roles = [4, [a0, a1, a2, a3], [a2, s]]

derived = [[2, 4, 5, 7], [], [[[a2, t], 6], [[s, a0], 4], [[s, a1], 5], [[s, a3], 7]], 22]

MFMC cut mask5 OUTWARD capacity := sum(4, 6, 5, 7)

MFMC cut mask5 OUTWARD capacity = 22

source roles = [5, [a0, a1, a2, a3], [a0, a2, s]]

derived = [[0, 2, 5, 7], [], [[[a0, t], 4], [[a2, t], 6], [[s, a1], 5], [[s, a3], 7]], 22]

MFMC cut mask6 OUTWARD capacity := sum(5, 6, 4, 7)

MFMC cut mask6 OUTWARD capacity = 22

source roles = [6, [a0, a1, a2, a3], [a1, a2, s]]

derived = [[1, 2, 4, 7], [], [[[a1, t], 5], [[a2, t], 6], [[s, a0], 4], [[s, a3], 7]], 22]

MFMC cut mask7 OUTWARD capacity := sum(4, 5, 6, 7)

MFMC cut mask7 OUTWARD capacity = 22

source roles = [7, [a0, a1, a2, a3], [a0, a1, a2, s]]

derived = [[0, 1, 2, 7], [], [[[a0, t], 4], [[a1, t], 5], [[a2, t], 6], [[s, a3], 7]], 22]

MFMC cut mask8 OUTWARD capacity := sum(7, 4, 5, 6)

MFMC cut mask8 OUTWARD capacity = 22

source roles = [8, [a0, a1, a2, a3], [a3, s]]

derived = [[3, 4, 5, 6], [], [[[a3, t], 7], [[s, a0], 4], [[s, a1], 5], [[s, a2], 6]], 22]

MFMC cut mask9 OUTWARD capacity := sum(4, 7, 5, 6)

MFMC cut mask9 OUTWARD capacity = 22

source roles = [9, [a0, a1, a2, a3], [a0, a3, s]]

derived = [[0, 3, 5, 6], [], [[[a0, t], 4], [[a3, t], 7], [[s, a1], 5], [[s, a2], 6]], 22]

MFMC cut mask10 OUTWARD capacity := sum(5, 7, 4, 6)

MFMC cut mask10 OUTWARD capacity = 22

source roles = [10, [a0, a1, a2, a3], [a1, a3, s]]

derived = [[1, 3, 4, 6], [], [[[a1, t], 5], [[a3, t], 7], [[s, a0], 4], [[s, a2], 6]], 22]

MFMC cut mask11 OUTWARD capacity := sum(4, 5, 7, 6)

MFMC cut mask11 OUTWARD capacity = 22

source roles = [11, [a0, a1, a2, a3], [a0, a1, a3, s]]

derived = [[0, 1, 3, 6], [], [[[a0, t], 4], [[a1, t], 5], [[a3, t], 7], [[s, a2], 6]], 22]

MFMC cut mask12 OUTWARD capacity := sum(6, 7, 4, 5)

MFMC cut mask12 OUTWARD capacity = 22

source roles = [12, [a0, a1, a2, a3], [a2, a3, s]]

derived = [[2, 3, 4, 5], [], [[[a2, t], 6], [[a3, t], 7], [[s, a0], 4], [[s, a1], 5]], 22]

MFMC cut mask13 OUTWARD capacity := sum(4, 6, 7, 5)

MFMC cut mask13 OUTWARD capacity = 22

source roles = [13, [a0, a1, a2, a3], [a0, a2, a3, s]]

derived = [[0, 2, 3, 5], [], [[[a0, t], 4], [[a2, t], 6], [[a3, t], 7], [[s, a1], 5]], 22]

MFMC cut mask14 OUTWARD capacity := sum(5, 6, 7, 4)

MFMC cut mask14 OUTWARD capacity = 22

source roles = [14, [a0, a1, a2, a3], [a1, a2, a3, s]]

derived = [[1, 2, 3, 4], [], [[[a1, t], 5], [[a2, t], 6], [[a3, t], 7], [[s, a0], 4]], 22]

MFMC cut mask15 OUTWARD capacity := sum(4, 5, 6, 7)

MFMC cut mask15 OUTWARD capacity = 22

source roles = [15, [a0, a1, a2, a3], [a0, a1, a2, a3, s]]

derived = [[0, 1, 2, 3], [], [[[a0, t], 4], [[a1, t], 5], [[a2, t], 6], [[a3, t], 7]], 22]

source roles = [[a0, a1, a2, a3, s, t], [a0, a1, a2, a3]]

derived = [16, 16, 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]]]

Reverse checks

• S035 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S034 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S033 · premise_and_result_audit — Recompute MFMC cut mask15 OUTWARD capacity with exact rational arithmetic, not rounded decimals.
• S032 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S031 · premise_and_result_audit — Recompute MFMC cut mask14 OUTWARD capacity with exact rational arithmetic, not rounded decimals.
• S030 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S029 · premise_and_result_audit — Recompute MFMC cut mask13 OUTWARD capacity with exact rational arithmetic, not rounded decimals.
• S028 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S027 · premise_and_result_audit — Recompute MFMC cut mask12 OUTWARD capacity with exact rational arithmetic, not rounded decimals.
• S026 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S025 · premise_and_result_audit — Recompute MFMC cut mask11 OUTWARD capacity with exact rational arithmetic, not rounded decimals.
• S024 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S023 · premise_and_result_audit — Recompute MFMC cut mask10 OUTWARD capacity with exact rational arithmetic, not rounded decimals.
• S022 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S021 · premise_and_result_audit — Recompute MFMC cut mask9 OUTWARD capacity with exact rational arithmetic, not rounded decimals.
• S020 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S019 · premise_and_result_audit — Recompute MFMC cut mask8 OUTWARD capacity with exact rational arithmetic, not rounded decimals.
• S018 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S017 · premise_and_result_audit — Recompute MFMC cut mask7 OUTWARD capacity with exact rational arithmetic, not rounded decimals.
• S016 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S015 · premise_and_result_audit — Recompute MFMC cut mask6 OUTWARD capacity with exact rational arithmetic, not rounded decimals.
• S014 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S013 · premise_and_result_audit — Recompute MFMC cut mask5 OUTWARD capacity with exact rational arithmetic, not rounded decimals.
• S012 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S011 · premise_and_result_audit — Recompute MFMC cut mask4 OUTWARD capacity with exact rational arithmetic, not rounded decimals.
• S010 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S009 · premise_and_result_audit — Recompute MFMC cut mask3 OUTWARD capacity with exact rational arithmetic, not rounded decimals.
• S008 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S007 · premise_and_result_audit — Recompute MFMC cut mask2 OUTWARD capacity with exact rational arithmetic, not rounded decimals.
• S006 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S005 · premise_and_result_audit — Recompute MFMC cut mask1 OUTWARD capacity with exact rational arithmetic, not rounded decimals.
• S004 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S003 · premise_and_result_audit — Recompute MFMC cut mask0 OUTWARD capacity with exact rational arithmetic, not rounded decimals.
• S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.

Result

([[[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)

The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.

This question review and source binding

Review date: 2026-10-26

Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof.