Question 25981: The Minimum Cut Is Selected from the Complete Cut Family

Original question and givens

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).

• G001 — 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).

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, t], [[s, a0], [a0, t], [s, a1], [a1, t], [s, a2], [a2, t]], [7, 7, 8, 8, 9, 9], null, s, t]

derived = [[a0, a1, a2, s, t], [[a0, t], [a1, t], [a2, t], [s, a0], [s, a1], [s, a2]], [7, 8, 9, 7, 8, 9], null]

MFMC cut mask0 OUTWARD capacity := sum(7, 8, 9)

MFMC cut mask0 OUTWARD capacity = 24

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

derived = [[3, 4, 5], [], [[[s, a0], 7], [[s, a1], 8], [[s, a2], 9]], 24]

MFMC cut mask1 OUTWARD capacity := sum(7, 8, 9)

MFMC cut mask1 OUTWARD capacity = 24

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

derived = [[0, 4, 5], [], [[[a0, t], 7], [[s, a1], 8], [[s, a2], 9]], 24]

MFMC cut mask2 OUTWARD capacity := sum(8, 7, 9)

MFMC cut mask2 OUTWARD capacity = 24

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

derived = [[1, 3, 5], [], [[[a1, t], 8], [[s, a0], 7], [[s, a2], 9]], 24]

MFMC cut mask3 OUTWARD capacity := sum(7, 8, 9)

MFMC cut mask3 OUTWARD capacity = 24

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

derived = [[0, 1, 5], [], [[[a0, t], 7], [[a1, t], 8], [[s, a2], 9]], 24]

MFMC cut mask4 OUTWARD capacity := sum(9, 7, 8)

MFMC cut mask4 OUTWARD capacity = 24

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

derived = [[2, 3, 4], [], [[[a2, t], 9], [[s, a0], 7], [[s, a1], 8]], 24]

MFMC cut mask5 OUTWARD capacity := sum(7, 9, 8)

MFMC cut mask5 OUTWARD capacity = 24

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

derived = [[0, 2, 4], [], [[[a0, t], 7], [[a2, t], 9], [[s, a1], 8]], 24]

MFMC cut mask6 OUTWARD capacity := sum(8, 9, 7)

MFMC cut mask6 OUTWARD capacity = 24

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

derived = [[1, 2, 3], [], [[[a1, t], 8], [[a2, t], 9], [[s, a0], 7]], 24]

MFMC cut mask7 OUTWARD capacity := sum(7, 8, 9)

MFMC cut mask7 OUTWARD capacity = 24

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

derived = [[0, 1, 2], [], [[[a0, t], 7], [[a1, t], 8], [[a2, t], 9]], 24]

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

derived = [8, 8, 24, [[s], [a0, s], [a1, s], [a0, a1, s], [a2, s], [a0, a2, s], [a1, a2, s], [a0, a1, a2, s]]]

Reverse checks

• S019 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• 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], 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)

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.