# 最大流最小割定理基礎 · 脈衝 R05

本批已通過 Schema、去重、來源快照與獨立答案重算；仍未標記為人工審定。

本題庫已有對應的中英逐步詳解。原題的 candidate 與英文 ai_candidate 狀態保留；AI 覆核、有限計算與 Lean 形式證明分開標示。

batch-20261015-advance-20261115-r05-max-flow-min-cut-theorem-foundations · 2026-10-15 · 100 題

欄序為來源綁定的呈現補充；原始 JSONL 與數學答案不變。

[欄序與來源綁定資料](https://ocme.evemisslab.com/data/i18n/question-output-protocols.json)



## 1. 最大流最小割定理

完整有限network、全部s–t cuts與forward-edge cut capacity。操作=mfmc-complete-cut-enumeration；資料={"vertices":["s","a0","a1","t"],"edges":[["s","a0"],["a0","t"],["s","a1"],["a1","t"],["a0","s"]],"source":"s","sink":"t","capacities":[2,2,3,3,7]}。請依 [vertices, source, sink, cut_rows, cut_count, required_cut_count, certificate] 欄序回答。

查看答案與解釋([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)

Wave6第九層核對=([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)；依完整明示資料獨立重算；feasible flow對每個cut皆有value≤capacity；同值flow/cut給雙optimality certificate；單一flow/cut不自動optimal；不計反向edges且不宣稱integrality、augmenting algorithm或一般定理已由instances證明。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r05-max-flow-min-cut-theorem-0001/)

最大流最小割定理 · qf-mod46-mfmc-complete-cut-enumeration · candidate





## 2. 最大流最小割定理

完整有限network、全部s–t cuts與forward-edge cut capacity。操作=mfmc-complete-cut-enumeration；資料={"vertices":["s","a0","a1","a2","t"],"edges":[["s","a0"],["a0","t"],["s","a1"],["a1","t"],["s","a2"],["a2","t"]],"source":"s","sink":"t","capacities":[3,3,4,4,5,5]}。請依 [vertices, source, sink, cut_rows, cut_count, required_cut_count, certificate] 欄序回答。

查看答案與解釋([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)

Wave6第九層核對=([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)；依完整明示資料獨立重算；feasible flow對每個cut皆有value≤capacity；同值flow/cut給雙optimality certificate；單一flow/cut不自動optimal；不計反向edges且不宣稱integrality、augmenting algorithm或一般定理已由instances證明。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r05-max-flow-min-cut-theorem-0002/)

最大流最小割定理 · qf-mod46-mfmc-complete-cut-enumeration · candidate





## 3. 最大流最小割定理

完整有限network、全部s–t cuts與forward-edge cut capacity。操作=mfmc-complete-cut-enumeration；資料={"vertices":["s","a0","a1","a2","a3","t"],"edges":[["s","a0"],["a0","t"],["s","a1"],["a1","t"],["s","a2"],["a2","t"],["s","a3"],["a3","t"]],"source":"s","sink":"t","capacities":[4,4,5,5,6,6,7,7]}。請依 [vertices, source, sink, cut_rows, cut_count, required_cut_count, certificate] 欄序回答。

查看答案與解釋([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)

Wave6第九層核對=([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)；依完整明示資料獨立重算；feasible flow對每個cut皆有value≤capacity；同值flow/cut給雙optimality certificate；單一flow/cut不自動optimal；不計反向edges且不宣稱integrality、augmenting algorithm或一般定理已由instances證明。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r05-max-flow-min-cut-theorem-0003/)

最大流最小割定理 · qf-mod46-mfmc-complete-cut-enumeration · candidate





## 4. 最大流最小割定理

完整有限network、全部s–t cuts與forward-edge cut capacity。操作=mfmc-complete-cut-enumeration；資料={"vertices":["s","a0","a1","a2","a3","a4","t"],"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","capacities":[5,5,6,6,7,7,8,8,9,9,10]}。請依 [vertices, source, sink, cut_rows, cut_count, required_cut_count, certificate] 欄序回答。

查看答案與解釋([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)

Wave6第九層核對=([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)；依完整明示資料獨立重算；feasible flow對每個cut皆有value≤capacity；同值flow/cut給雙optimality certificate；單一flow/cut不自動optimal；不計反向edges且不宣稱integrality、augmenting algorithm或一般定理已由instances證明。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r05-max-flow-min-cut-theorem-0004/)

最大流最小割定理 · qf-mod46-mfmc-complete-cut-enumeration · candidate





## 5. 最大流最小割定理

完整有限network、全部s–t cuts與forward-edge cut capacity。操作=mfmc-complete-cut-enumeration；資料={"vertices":["s","a0","a1","t"],"edges":[["s","a0"],["a0","t"],["s","a1"],["a1","t"]],"source":"s","sink":"t","capacities":[6,6,7,7]}。請依 [vertices, source, sink, cut_rows, cut_count, required_cut_count, certificate] 欄序回答。

查看答案與解釋([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)

Wave6第九層核對=([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)；依完整明示資料獨立重算；feasible flow對每個cut皆有value≤capacity；同值flow/cut給雙optimality certificate；單一flow/cut不自動optimal；不計反向edges且不宣稱integrality、augmenting algorithm或一般定理已由instances證明。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r05-max-flow-min-cut-theorem-0005/)

最大流最小割定理 · qf-mod46-mfmc-complete-cut-enumeration · candidate





## 6. 最大流最小割定理

完整有限network、全部s–t cuts與forward-edge cut capacity。操作=mfmc-complete-cut-enumeration；資料={"vertices":["s","a0","a1","a2","t"],"edges":[["s","a0"],["a0","t"],["s","a1"],["a1","t"],["s","a2"],["a2","t"]],"source":"s","sink":"t","capacities":[7,7,8,8,9,9]}。請依 [vertices, source, sink, cut_rows, cut_count, required_cut_count, certificate] 欄序回答。

查看答案與解釋([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)

Wave6第九層核對=([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)；依完整明示資料獨立重算；feasible flow對每個cut皆有value≤capacity；同值flow/cut給雙optimality certificate；單一flow/cut不自動optimal；不計反向edges且不宣稱integrality、augmenting algorithm或一般定理已由instances證明。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r05-max-flow-min-cut-theorem-0006/)

最大流最小割定理 · qf-mod46-mfmc-complete-cut-enumeration · candidate





## 7. 最大流最小割定理

完整有限network、全部s–t cuts與forward-edge cut capacity。操作=mfmc-complete-cut-enumeration；資料={"vertices":["s","a0","a1","a2","a3","t"],"edges":[["s","a0"],["a0","t"],["s","a1"],["a1","t"],["s","a2"],["a2","t"],["s","a3"],["a3","t"],["a0","s"]],"source":"s","sink":"t","capacities":[8,8,9,9,10,10,11,11,13]}。請依 [vertices, source, sink, cut_rows, cut_count, required_cut_count, certificate] 欄序回答。

查看答案與解釋([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)

Wave6第九層核對=([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)；依完整明示資料獨立重算；feasible flow對每個cut皆有value≤capacity；同值flow/cut給雙optimality certificate；單一flow/cut不自動optimal；不計反向edges且不宣稱integrality、augmenting algorithm或一般定理已由instances證明。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r05-max-flow-min-cut-theorem-0007/)

最大流最小割定理 · qf-mod46-mfmc-complete-cut-enumeration · candidate





## 8. 最大流最小割定理

完整有限network、全部s–t cuts與forward-edge cut capacity。操作=mfmc-complete-cut-enumeration；資料={"vertices":["s","a0","a1","a2","a3","a4","t"],"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","capacities":[9,9,10,10,11,11,12,12,13,13]}。請依 [vertices, source, sink, cut_rows, cut_count, required_cut_count, certificate] 欄序回答。

查看答案與解釋([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)

Wave6第九層核對=([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)；依完整明示資料獨立重算；feasible flow對每個cut皆有value≤capacity；同值flow/cut給雙optimality certificate；單一flow/cut不自動optimal；不計反向edges且不宣稱integrality、augmenting algorithm或一般定理已由instances證明。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r05-max-flow-min-cut-theorem-0008/)

最大流最小割定理 · qf-mod46-mfmc-complete-cut-enumeration · candidate





## 9. 最大流最小割定理

完整有限network、全部s–t cuts與forward-edge cut capacity。操作=mfmc-complete-cut-enumeration；資料={"vertices":["s","a0","a1","t"],"edges":[["s","a0"],["a0","t"],["s","a1"],["a1","t"]],"source":"s","sink":"t","capacities":[10,10,11,11]}。請依 [vertices, source, sink, cut_rows, cut_count, required_cut_count, certificate] 欄序回答。

查看答案與解釋([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)

Wave6第九層核對=([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)；依完整明示資料獨立重算；feasible flow對每個cut皆有value≤capacity；同值flow/cut給雙optimality certificate；單一flow/cut不自動optimal；不計反向edges且不宣稱integrality、augmenting algorithm或一般定理已由instances證明。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r05-max-flow-min-cut-theorem-0009/)

最大流最小割定理 · qf-mod46-mfmc-complete-cut-enumeration · candidate





## 10. 最大流最小割定理

完整有限network、全部s–t cuts與forward-edge cut capacity。操作=mfmc-complete-cut-enumeration；資料={"vertices":["s","a0","a1","a2","t"],"edges":[["s","a0"],["a0","t"],["s","a1"],["a1","t"],["s","a2"],["a2","t"],["a0","s"]],"source":"s","sink":"t","capacities":[11,11,12,12,13,13,16]}。請依 [vertices, source, sink, cut_rows, cut_count, required_cut_count, certificate] 欄序回答。

查看答案與解釋([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)

Wave6第九層核對=([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)；依完整明示資料獨立重算；feasible flow對每個cut皆有value≤capacity；同值flow/cut給雙optimality certificate；單一flow/cut不自動optimal；不計反向edges且不宣稱integrality、augmenting algorithm或一般定理已由instances證明。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r05-max-flow-min-cut-theorem-0010/)

最大流最小割定理 · qf-mod46-mfmc-complete-cut-enumeration · candidate





## 11. 最大流最小割定理

完整有限network、全部s–t cuts與forward-edge cut capacity。操作=mfmc-complete-cut-enumeration；資料={"vertices":["s","a0","a1","a2","a3","t"],"edges":[["s","a0"],["a0","t"],["s","a1"],["a1","t"],["s","a2"],["a2","t"],["s","a3"],["a3","t"]],"source":"s","sink":"t","capacities":[12,12,13,13,14,14,15,15]}。請依 [vertices, source, sink, cut_rows, cut_count, required_cut_count, certificate] 欄序回答。

查看答案與解釋([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)

Wave6第九層核對=([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)；依完整明示資料獨立重算；feasible flow對每個cut皆有value≤capacity；同值flow/cut給雙optimality certificate；單一flow/cut不自動optimal；不計反向edges且不宣稱integrality、augmenting algorithm或一般定理已由instances證明。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r05-max-flow-min-cut-theorem-0011/)

最大流最小割定理 · qf-mod46-mfmc-complete-cut-enumeration · candidate





## 12. 最大流最小割定理

完整有限network、全部s–t cuts與forward-edge cut capacity。操作=mfmc-complete-cut-enumeration；資料={"vertices":["s","a0","a1","a2","a3","a4","t"],"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","capacities":[13,13,14,14,15,15,16,16,17,17]}。請依 [vertices, source, sink, cut_rows, cut_count, required_cut_count, certificate] 欄序回答。

查看答案與解釋([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)

Wave6第九層核對=([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)；依完整明示資料獨立重算；feasible flow對每個cut皆有value≤capacity；同值flow/cut給雙optimality certificate；單一flow/cut不自動optimal；不計反向edges且不宣稱integrality、augmenting algorithm或一般定理已由instances證明。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r05-max-flow-min-cut-theorem-0012/)

最大流最小割定理 · qf-mod46-mfmc-complete-cut-enumeration · candidate





## 13. 最大流最小割定理

完整有限network、全部s–t cuts與forward-edge cut capacity。操作=mfmc-complete-cut-enumeration；資料={"vertices":["s","a0","a1","t"],"edges":[["s","a0"],["a0","t"],["s","a1"],["a1","t"],["a0","s"]],"source":"s","sink":"t","capacities":[14,14,15,15,19]}。請依 [vertices, source, sink, cut_rows, cut_count, required_cut_count, certificate] 欄序回答。

查看答案與解釋([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)

Wave6第九層核對=([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)；依完整明示資料獨立重算；feasible flow對每個cut皆有value≤capacity；同值flow/cut給雙optimality certificate；單一flow/cut不自動optimal；不計反向edges且不宣稱integrality、augmenting algorithm或一般定理已由instances證明。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r05-max-flow-min-cut-theorem-0013/)

最大流最小割定理 · qf-mod46-mfmc-complete-cut-enumeration · candidate





## 14. 最大流最小割定理

完整有限network、全部s–t cuts與forward-edge cut capacity。操作=mfmc-complete-cut-enumeration；資料={"vertices":["s","a0","a1","a2","t"],"edges":[["s","a0"],["a0","t"],["s","a1"],["a1","t"],["s","a2"],["a2","t"]],"source":"s","sink":"t","capacities":[15,15,16,16,17,17]}。請依 [vertices, source, sink, cut_rows, cut_count, required_cut_count, certificate] 欄序回答。

查看答案與解釋([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)

Wave6第九層核對=([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)；依完整明示資料獨立重算；feasible flow對每個cut皆有value≤capacity；同值flow/cut給雙optimality certificate；單一flow/cut不自動optimal；不計反向edges且不宣稱integrality、augmenting algorithm或一般定理已由instances證明。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r05-max-flow-min-cut-theorem-0014/)

最大流最小割定理 · qf-mod46-mfmc-complete-cut-enumeration · candidate





## 15. 最大流最小割定理

完整有限network、全部s–t cuts與forward-edge cut capacity。操作=mfmc-complete-cut-enumeration；資料={"vertices":["s","a0","a1","a2","a3","t"],"edges":[["s","a0"],["a0","t"],["s","a1"],["a1","t"],["s","a2"],["a2","t"],["s","a3"],["a3","t"]],"source":"s","sink":"t","capacities":[16,16,17,17,18,18,19,19]}。請依 [vertices, source, sink, cut_rows, cut_count, required_cut_count, certificate] 欄序回答。

查看答案與解釋([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)

Wave6第九層核對=([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)；依完整明示資料獨立重算；feasible flow對每個cut皆有value≤capacity；同值flow/cut給雙optimality certificate；單一flow/cut不自動optimal；不計反向edges且不宣稱integrality、augmenting algorithm或一般定理已由instances證明。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r05-max-flow-min-cut-theorem-0015/)

最大流最小割定理 · qf-mod46-mfmc-complete-cut-enumeration · candidate





## 16. 最大流最小割定理

完整有限network、全部s–t cuts與forward-edge cut capacity。操作=mfmc-complete-cut-enumeration；資料={"vertices":["s","a0","a1","a2","a3","a4","t"],"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","capacities":[17,17,18,18,19,19,20,20,21,21,22]}。請依 [vertices, source, sink, cut_rows, cut_count, required_cut_count, certificate] 欄序回答。

查看答案與解釋([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)

Wave6第九層核對=([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)；依完整明示資料獨立重算；feasible flow對每個cut皆有value≤capacity；同值flow/cut給雙optimality certificate；單一flow/cut不自動optimal；不計反向edges且不宣稱integrality、augmenting algorithm或一般定理已由instances證明。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r05-max-flow-min-cut-theorem-0016/)

最大流最小割定理 · qf-mod46-mfmc-complete-cut-enumeration · candidate





## 17. 最大流最小割定理

完整有限network、全部s–t cuts與forward-edge cut capacity。操作=mfmc-complete-cut-enumeration；資料={"vertices":["s","a0","a1","t"],"edges":[["s","a0"],["a0","t"],["s","a1"],["a1","t"]],"source":"s","sink":"t","capacities":[18,18,19,19]}。請依 [vertices, source, sink, cut_rows, cut_count, required_cut_count, certificate] 欄序回答。

查看答案與解釋([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)

Wave6第九層核對=([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)；依完整明示資料獨立重算；feasible flow對每個cut皆有value≤capacity；同值flow/cut給雙optimality certificate；單一flow/cut不自動optimal；不計反向edges且不宣稱integrality、augmenting algorithm或一般定理已由instances證明。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r05-max-flow-min-cut-theorem-0017/)

最大流最小割定理 · qf-mod46-mfmc-complete-cut-enumeration · candidate





## 18. 最大流最小割定理

完整有限network、全部s–t cuts與forward-edge cut capacity。操作=mfmc-complete-cut-enumeration；資料={"vertices":["s","a0","a1","a2","t"],"edges":[["s","a0"],["a0","t"],["s","a1"],["a1","t"],["s","a2"],["a2","t"]],"source":"s","sink":"t","capacities":[19,19,20,20,21,21]}。請依 [vertices, source, sink, cut_rows, cut_count, required_cut_count, certificate] 欄序回答。

查看答案與解釋([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)

Wave6第九層核對=([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)；依完整明示資料獨立重算；feasible flow對每個cut皆有value≤capacity；同值flow/cut給雙optimality certificate；單一flow/cut不自動optimal；不計反向edges且不宣稱integrality、augmenting algorithm或一般定理已由instances證明。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r05-max-flow-min-cut-theorem-0018/)

最大流最小割定理 · qf-mod46-mfmc-complete-cut-enumeration · candidate





## 19. 最大流最小割定理

完整有限network、全部s–t cuts與forward-edge cut capacity。操作=mfmc-complete-cut-enumeration；資料={"vertices":["s","a0","a1","a2","a3","t"],"edges":[["s","a0"],["a0","t"],["s","a1"],["a1","t"],["s","a2"],["a2","t"],["s","a3"],["a3","t"],["a0","s"]],"source":"s","sink":"t","capacities":[20,20,21,21,22,22,23,23,25]}。請依 [vertices, source, sink, cut_rows, cut_count, required_cut_count, certificate] 欄序回答。

查看答案與解釋([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)

Wave6第九層核對=([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)；依完整明示資料獨立重算；feasible flow對每個cut皆有value≤capacity；同值flow/cut給雙optimality certificate；單一flow/cut不自動optimal；不計反向edges且不宣稱integrality、augmenting algorithm或一般定理已由instances證明。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r05-max-flow-min-cut-theorem-0019/)

最大流最小割定理 · qf-mod46-mfmc-complete-cut-enumeration · candidate





## 20. 最大流最小割定理

完整有限network、全部s–t cuts與forward-edge cut capacity。操作=mfmc-complete-cut-enumeration；資料={"vertices":["s","a0","a1","a2","a3","a4","t"],"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","capacities":[21,21,22,22,23,23,24,24,25,25]}。請依 [vertices, source, sink, cut_rows, cut_count, required_cut_count, certificate] 欄序回答。

查看答案與解釋([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)

Wave6第九層核對=([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)；依完整明示資料獨立重算；feasible flow對每個cut皆有value≤capacity；同值flow/cut給雙optimality certificate；單一flow/cut不自動optimal；不計反向edges且不宣稱integrality、augmenting algorithm或一般定理已由instances證明。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r05-max-flow-min-cut-theorem-0020/)

最大流最小割定理 · qf-mod46-mfmc-complete-cut-enumeration · candidate





## 21. 最大流最小割定理

完整有限network、全部s–t cuts與forward-edge cut capacity。操作=mfmc-complete-cut-enumeration；資料={"vertices":["s","a0","a1","t"],"edges":[["s","a0"],["a0","t"],["s","a1"],["a1","t"]],"source":"s","sink":"t","capacities":[22,22,23,23]}。請依 [vertices, source, sink, cut_rows, cut_count, required_cut_count, certificate] 欄序回答。

查看答案與解釋([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)

Wave6第九層核對=([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)；依完整明示資料獨立重算；feasible flow對每個cut皆有value≤capacity；同值flow/cut給雙optimality certificate；單一flow/cut不自動optimal；不計反向edges且不宣稱integrality、augmenting algorithm或一般定理已由instances證明。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r05-max-flow-min-cut-theorem-0021/)

最大流最小割定理 · qf-mod46-mfmc-complete-cut-enumeration · candidate





## 22. 最大流最小割定理

完整有限network、全部s–t cuts與forward-edge cut capacity。操作=mfmc-complete-cut-enumeration；資料={"vertices":["s","a0","a1","a2","t"],"edges":[["s","a0"],["a0","t"],["s","a1"],["a1","t"],["s","a2"],["a2","t"],["a0","s"]],"source":"s","sink":"t","capacities":[23,23,24,24,25,25,28]}。請依 [vertices, source, sink, cut_rows, cut_count, required_cut_count, certificate] 欄序回答。

查看答案與解釋([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)

Wave6第九層核對=([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)；依完整明示資料獨立重算；feasible flow對每個cut皆有value≤capacity；同值flow/cut給雙optimality certificate；單一flow/cut不自動optimal；不計反向edges且不宣稱integrality、augmenting algorithm或一般定理已由instances證明。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r05-max-flow-min-cut-theorem-0022/)

最大流最小割定理 · qf-mod46-mfmc-complete-cut-enumeration · candidate





## 23. 最大流最小割定理

完整有限network、全部s–t cuts與forward-edge cut capacity。操作=mfmc-complete-cut-enumeration；資料={"vertices":["s","a0","a1","a2","a3","t"],"edges":[["s","a0"],["a0","t"],["s","a1"],["a1","t"],["s","a2"],["a2","t"],["s","a3"],["a3","t"]],"source":"s","sink":"t","capacities":[24,24,25,25,26,26,27,27]}。請依 [vertices, source, sink, cut_rows, cut_count, required_cut_count, certificate] 欄序回答。

查看答案與解釋([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)

Wave6第九層核對=([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)；依完整明示資料獨立重算；feasible flow對每個cut皆有value≤capacity；同值flow/cut給雙optimality certificate；單一flow/cut不自動optimal；不計反向edges且不宣稱integrality、augmenting algorithm或一般定理已由instances證明。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r05-max-flow-min-cut-theorem-0023/)

最大流最小割定理 · qf-mod46-mfmc-complete-cut-enumeration · candidate





## 24. 最大流最小割定理

完整有限network、全部s–t cuts與forward-edge cut capacity。操作=mfmc-complete-cut-enumeration；資料={"vertices":["s","a0","a1","a2","a3","a4","t"],"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","capacities":[25,25,26,26,27,27,28,28,29,29]}。請依 [vertices, source, sink, cut_rows, cut_count, required_cut_count, certificate] 欄序回答。

查看答案與解釋([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)

Wave6第九層核對=([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)；依完整明示資料獨立重算；feasible flow對每個cut皆有value≤capacity；同值flow/cut給雙optimality certificate；單一flow/cut不自動optimal；不計反向edges且不宣稱integrality、augmenting algorithm或一般定理已由instances證明。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r05-max-flow-min-cut-theorem-0024/)

最大流最小割定理 · qf-mod46-mfmc-complete-cut-enumeration · candidate





## 25. 最大流最小割定理

完整有限network、全部s–t cuts與forward-edge cut capacity。操作=mfmc-complete-cut-enumeration；資料={"vertices":["s","a0","a1","t"],"edges":[["s","a0"],["a0","t"],["s","a1"],["a1","t"],["a0","s"]],"source":"s","sink":"t","capacities":[26,26,27,27,31]}。請依 [vertices, source, sink, cut_rows, cut_count, required_cut_count, certificate] 欄序回答。

查看答案與解釋([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)

Wave6第九層核對=([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)；依完整明示資料獨立重算；feasible flow對每個cut皆有value≤capacity；同值flow/cut給雙optimality certificate；單一flow/cut不自動optimal；不計反向edges且不宣稱integrality、augmenting algorithm或一般定理已由instances證明。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r05-max-flow-min-cut-theorem-0025/)

最大流最小割定理 · qf-mod46-mfmc-complete-cut-enumeration · candidate





## 26. 最大流最小割定理

完整有限network、全部s–t cuts與forward-edge cut capacity。操作=mfmc-flow-cut-upper-bound；資料={"vertices":["s","a0","a1","t"],"edges":[["s","a0"],["a0","t"],["s","a1"],["a1","t"],["a0","s"]],"source":"s","sink":"t","capacities":[2,2,3,3,7],"flows":[1,1,2,2,0]}。請依 [source_value, sink_value, flow_cut_rows, all_cut_bounds_hold, automatic_maximum_claim, certificate] 欄序回答。

cut capacity只算向外邊；flow net是向外flow減向內flow。automatic_maximum_claim的否不是宣稱這個flow一定非最大。

查看答案與解釋(3, 3, [[[s], 3, 0, 3, 5, 是, 是], [[a0, s], 3, 0, 3, 5, 是, 是], [[a1, s], 3, 0, 3, 5, 是, 是], [[a0, a1, s], 3, 0, 3, 5, 是, 是]], 是, 否, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum)

Wave6第九層核對=(3, 3, [[[s], 3, 0, 3, 5, 是, 是], [[a0, s], 3, 0, 3, 5, 是, 是], [[a1, s], 3, 0, 3, 5, 是, 是], [[a0, a1, s], 3, 0, 3, 5, 是, 是]], 是, 否, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum)；依完整明示資料獨立重算；feasible flow對每個cut皆有value≤capacity；同值flow/cut給雙optimality certificate；單一flow/cut不自動optimal；不計反向edges且不宣稱integrality、augmenting algorithm或一般定理已由instances證明。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r05-max-flow-min-cut-theorem-0026/)

最大流最小割定理 · qf-mod46-mfmc-flow-cut-upper-bound · candidate





## 27. 最大流最小割定理

完整有限network、全部s–t cuts與forward-edge cut capacity。操作=mfmc-flow-cut-upper-bound；資料={"vertices":["s","a0","a1","a2","t"],"edges":[["s","a0"],["a0","t"],["s","a1"],["a1","t"],["s","a2"],["a2","t"]],"source":"s","sink":"t","capacities":[3,3,4,4,5,5],"flows":[2,2,3,3,4,4]}。請依 [source_value, sink_value, flow_cut_rows, all_cut_bounds_hold, automatic_maximum_claim, certificate] 欄序回答。

cut capacity只算向外邊；flow net是向外flow減向內flow。automatic_maximum_claim的否不是宣稱這個flow一定非最大。

查看答案與解釋(9, 9, [[[s], 9, 0, 9, 12, 是, 是], [[a0, s], 9, 0, 9, 12, 是, 是], [[a1, s], 9, 0, 9, 12, 是, 是], [[a0, a1, s], 9, 0, 9, 12, 是, 是], [[a2, s], 9, 0, 9, 12, 是, 是], [[a0, a2, s], 9, 0, 9, 12, 是, 是], [[a1, a2, s], 9, 0, 9, 12, 是, 是], [[a0, a1, a2, s], 9, 0, 9, 12, 是, 是]], 是, 否, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum)

Wave6第九層核對=(9, 9, [[[s], 9, 0, 9, 12, 是, 是], [[a0, s], 9, 0, 9, 12, 是, 是], [[a1, s], 9, 0, 9, 12, 是, 是], [[a0, a1, s], 9, 0, 9, 12, 是, 是], [[a2, s], 9, 0, 9, 12, 是, 是], [[a0, a2, s], 9, 0, 9, 12, 是, 是], [[a1, a2, s], 9, 0, 9, 12, 是, 是], [[a0, a1, a2, s], 9, 0, 9, 12, 是, 是]], 是, 否, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum)；依完整明示資料獨立重算；feasible flow對每個cut皆有value≤capacity；同值flow/cut給雙optimality certificate；單一flow/cut不自動optimal；不計反向edges且不宣稱integrality、augmenting algorithm或一般定理已由instances證明。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r05-max-flow-min-cut-theorem-0027/)

最大流最小割定理 · qf-mod46-mfmc-flow-cut-upper-bound · candidate





## 28. 最大流最小割定理

完整有限network、全部s–t cuts與forward-edge cut capacity。操作=mfmc-flow-cut-upper-bound；資料={"vertices":["s","a0","a1","a2","a3","t"],"edges":[["s","a0"],["a0","t"],["s","a1"],["a1","t"],["s","a2"],["a2","t"],["s","a3"],["a3","t"]],"source":"s","sink":"t","capacities":[4,4,5,5,6,6,7,7],"flows":[3,3,4,4,5,5,6,6]}。請依 [source_value, sink_value, flow_cut_rows, all_cut_bounds_hold, automatic_maximum_claim, certificate] 欄序回答。

cut capacity只算向外邊；flow net是向外flow減向內flow。automatic_maximum_claim的否不是宣稱這個flow一定非最大。

查看答案與解釋(18, 18, [[[s], 18, 0, 18, 22, 是, 是], [[a0, s], 18, 0, 18, 22, 是, 是], [[a1, s], 18, 0, 18, 22, 是, 是], [[a0, a1, s], 18, 0, 18, 22, 是, 是], [[a2, s], 18, 0, 18, 22, 是, 是], [[a0, a2, s], 18, 0, 18, 22, 是, 是], [[a1, a2, s], 18, 0, 18, 22, 是, 是], [[a0, a1, a2, s], 18, 0, 18, 22, 是, 是], [[a3, s], 18, 0, 18, 22, 是, 是], [[a0, a3, s], 18, 0, 18, 22, 是, 是], [[a1, a3, s], 18, 0, 18, 22, 是, 是], [[a0, a1, a3, s], 18, 0, 18, 22, 是, 是], [[a2, a3, s], 18, 0, 18, 22, 是, 是], [[a0, a2, a3, s], 18, 0, 18, 22, 是, 是], [[a1, a2, a3, s], 18, 0, 18, 22, 是, 是], [[a0, a1, a2, a3, s], 18, 0, 18, 22, 是, 是]], 是, 否, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum)

Wave6第九層核對=(18, 18, [[[s], 18, 0, 18, 22, 是, 是], [[a0, s], 18, 0, 18, 22, 是, 是], [[a1, s], 18, 0, 18, 22, 是, 是], [[a0, a1, s], 18, 0, 18, 22, 是, 是], [[a2, s], 18, 0, 18, 22, 是, 是], [[a0, a2, s], 18, 0, 18, 22, 是, 是], [[a1, a2, s], 18, 0, 18, 22, 是, 是], [[a0, a1, a2, s], 18, 0, 18, 22, 是, 是], [[a3, s], 18, 0, 18, 22, 是, 是], [[a0, a3, s], 18, 0, 18, 22, 是, 是], [[a1, a3, s], 18, 0, 18, 22, 是, 是], [[a0, a1, a3, s], 18, 0, 18, 22, 是, 是], [[a2, a3, s], 18, 0, 18, 22, 是, 是], [[a0, a2, a3, s], 18, 0, 18, 22, 是, 是], [[a1, a2, a3, s], 18, 0, 18, 22, 是, 是], [[a0, a1, a2, a3, s], 18, 0, 18, 22, 是, 是]], 是, 否, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum)；依完整明示資料獨立重算；feasible flow對每個cut皆有value≤capacity；同值flow/cut給雙optimality certificate；單一flow/cut不自動optimal；不計反向edges且不宣稱integrality、augmenting algorithm或一般定理已由instances證明。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r05-max-flow-min-cut-theorem-0028/)

最大流最小割定理 · qf-mod46-mfmc-flow-cut-upper-bound · candidate





## 29. 最大流最小割定理

完整有限network、全部s–t cuts與forward-edge cut capacity。操作=mfmc-flow-cut-upper-bound；資料={"vertices":["s","a0","a1","a2","a3","a4","t"],"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","capacities":[5,5,6,6,7,7,8,8,9,9,10],"flows":[4,4,5,5,6,6,7,7,8,8,0]}。請依 [source_value, sink_value, flow_cut_rows, all_cut_bounds_hold, automatic_maximum_claim, certificate] 欄序回答。

cut capacity只算向外邊；flow net是向外flow減向內flow。automatic_maximum_claim的否不是宣稱這個flow一定非最大。

查看答案與解釋(30, 30, [[[s], 30, 0, 30, 35, 是, 是], [[a0, s], 30, 0, 30, 35, 是, 是], [[a1, s], 30, 0, 30, 35, 是, 是], [[a0, a1, s], 30, 0, 30, 35, 是, 是], [[a2, s], 30, 0, 30, 35, 是, 是], [[a0, a2, s], 30, 0, 30, 35, 是, 是], [[a1, a2, s], 30, 0, 30, 35, 是, 是], [[a0, a1, a2, s], 30, 0, 30, 35, 是, 是], [[a3, s], 30, 0, 30, 35, 是, 是], [[a0, a3, s], 30, 0, 30, 35, 是, 是], [[a1, a3, s], 30, 0, 30, 35, 是, 是], [[a0, a1, a3, s], 30, 0, 30, 35, 是, 是], [[a2, a3, s], 30, 0, 30, 35, 是, 是], [[a0, a2, a3, s], 30, 0, 30, 35, 是, 是], [[a1, a2, a3, s], 30, 0, 30, 35, 是, 是], [[a0, a1, a2, a3, s], 30, 0, 30, 35, 是, 是], [[a4, s], 30, 0, 30, 35, 是, 是], [[a0, a4, s], 30, 0, 30, 35, 是, 是], [[a1, a4, s], 30, 0, 30, 35, 是, 是], [[a0, a1, a4, s], 30, 0, 30, 35, 是, 是], [[a2, a4, s], 30, 0, 30, 35, 是, 是], [[a0, a2, a4, s], 30, 0, 30, 35, 是, 是], [[a1, a2, a4, s], 30, 0, 30, 35, 是, 是], [[a0, a1, a2, a4, s], 30, 0, 30, 35, 是, 是], [[a3, a4, s], 30, 0, 30, 35, 是, 是], [[a0, a3, a4, s], 30, 0, 30, 35, 是, 是], [[a1, a3, a4, s], 30, 0, 30, 35, 是, 是], [[a0, a1, a3, a4, s], 30, 0, 30, 35, 是, 是], [[a2, a3, a4, s], 30, 0, 30, 35, 是, 是], [[a0, a2, a3, a4, s], 30, 0, 30, 35, 是, 是], [[a1, a2, a3, a4, s], 30, 0, 30, 35, 是, 是], [[a0, a1, a2, a3, a4, s], 30, 0, 30, 35, 是, 是]], 是, 否, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum)

Wave6第九層核對=(30, 30, [[[s], 30, 0, 30, 35, 是, 是], [[a0, s], 30, 0, 30, 35, 是, 是], [[a1, s], 30, 0, 30, 35, 是, 是], [[a0, a1, s], 30, 0, 30, 35, 是, 是], [[a2, s], 30, 0, 30, 35, 是, 是], [[a0, a2, s], 30, 0, 30, 35, 是, 是], [[a1, a2, s], 30, 0, 30, 35, 是, 是], [[a0, a1, a2, s], 30, 0, 30, 35, 是, 是], [[a3, s], 30, 0, 30, 35, 是, 是], [[a0, a3, s], 30, 0, 30, 35, 是, 是], [[a1, a3, s], 30, 0, 30, 35, 是, 是], [[a0, a1, a3, s], 30, 0, 30, 35, 是, 是], [[a2, a3, s], 30, 0, 30, 35, 是, 是], [[a0, a2, a3, s], 30, 0, 30, 35, 是, 是], [[a1, a2, a3, s], 30, 0, 30, 35, 是, 是], [[a0, a1, a2, a3, s], 30, 0, 30, 35, 是, 是], [[a4, s], 30, 0, 30, 35, 是, 是], [[a0, a4, s], 30, 0, 30, 35, 是, 是], [[a1, a4, s], 30, 0, 30, 35, 是, 是], [[a0, a1, a4, s], 30, 0, 30, 35, 是, 是], [[a2, a4, s], 30, 0, 30, 35, 是, 是], [[a0, a2, a4, s], 30, 0, 30, 35, 是, 是], [[a1, a2, a4, s], 30, 0, 30, 35, 是, 是], [[a0, a1, a2, a4, s], 30, 0, 30, 35, 是, 是], [[a3, a4, s], 30, 0, 30, 35, 是, 是], [[a0, a3, a4, s], 30, 0, 30, 35, 是, 是], [[a1, a3, a4, s], 30, 0, 30, 35, 是, 是], [[a0, a1, a3, a4, s], 30, 0, 30, 35, 是, 是], [[a2, a3, a4, s], 30, 0, 30, 35, 是, 是], [[a0, a2, a3, a4, s], 30, 0, 30, 35, 是, 是], [[a1, a2, a3, a4, s], 30, 0, 30, 35, 是, 是], [[a0, a1, a2, a3, a4, s], 30, 0, 30, 35, 是, 是]], 是, 否, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum)；依完整明示資料獨立重算；feasible flow對每個cut皆有value≤capacity；同值flow/cut給雙optimality certificate；單一flow/cut不自動optimal；不計反向edges且不宣稱integrality、augmenting algorithm或一般定理已由instances證明。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r05-max-flow-min-cut-theorem-0029/)

最大流最小割定理 · qf-mod46-mfmc-flow-cut-upper-bound · candidate





## 30. 最大流最小割定理

完整有限network、全部s–t cuts與forward-edge cut capacity。操作=mfmc-flow-cut-upper-bound；資料={"vertices":["s","a0","a1","t"],"edges":[["s","a0"],["a0","t"],["s","a1"],["a1","t"]],"source":"s","sink":"t","capacities":[6,6,7,7],"flows":[5,5,6,6]}。請依 [source_value, sink_value, flow_cut_rows, all_cut_bounds_hold, automatic_maximum_claim, certificate] 欄序回答。

cut capacity只算向外邊；flow net是向外flow減向內flow。automatic_maximum_claim的否不是宣稱這個flow一定非最大。

查看答案與解釋(11, 11, [[[s], 11, 0, 11, 13, 是, 是], [[a0, s], 11, 0, 11, 13, 是, 是], [[a1, s], 11, 0, 11, 13, 是, 是], [[a0, a1, s], 11, 0, 11, 13, 是, 是]], 是, 否, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum)

Wave6第九層核對=(11, 11, [[[s], 11, 0, 11, 13, 是, 是], [[a0, s], 11, 0, 11, 13, 是, 是], [[a1, s], 11, 0, 11, 13, 是, 是], [[a0, a1, s], 11, 0, 11, 13, 是, 是]], 是, 否, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum)；依完整明示資料獨立重算；feasible flow對每個cut皆有value≤capacity；同值flow/cut給雙optimality certificate；單一flow/cut不自動optimal；不計反向edges且不宣稱integrality、augmenting algorithm或一般定理已由instances證明。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r05-max-flow-min-cut-theorem-0030/)

最大流最小割定理 · qf-mod46-mfmc-flow-cut-upper-bound · candidate





## 31. 最大流最小割定理

完整有限network、全部s–t cuts與forward-edge cut capacity。操作=mfmc-flow-cut-upper-bound；資料={"vertices":["s","a0","a1","a2","t"],"edges":[["s","a0"],["a0","t"],["s","a1"],["a1","t"],["s","a2"],["a2","t"]],"source":"s","sink":"t","capacities":[7,7,8,8,9,9],"flows":[6,6,7,7,8,8]}。請依 [source_value, sink_value, flow_cut_rows, all_cut_bounds_hold, automatic_maximum_claim, certificate] 欄序回答。

cut capacity只算向外邊；flow net是向外flow減向內flow。automatic_maximum_claim的否不是宣稱這個flow一定非最大。

查看答案與解釋(21, 21, [[[s], 21, 0, 21, 24, 是, 是], [[a0, s], 21, 0, 21, 24, 是, 是], [[a1, s], 21, 0, 21, 24, 是, 是], [[a0, a1, s], 21, 0, 21, 24, 是, 是], [[a2, s], 21, 0, 21, 24, 是, 是], [[a0, a2, s], 21, 0, 21, 24, 是, 是], [[a1, a2, s], 21, 0, 21, 24, 是, 是], [[a0, a1, a2, s], 21, 0, 21, 24, 是, 是]], 是, 否, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum)

Wave6第九層核對=(21, 21, [[[s], 21, 0, 21, 24, 是, 是], [[a0, s], 21, 0, 21, 24, 是, 是], [[a1, s], 21, 0, 21, 24, 是, 是], [[a0, a1, s], 21, 0, 21, 24, 是, 是], [[a2, s], 21, 0, 21, 24, 是, 是], [[a0, a2, s], 21, 0, 21, 24, 是, 是], [[a1, a2, s], 21, 0, 21, 24, 是, 是], [[a0, a1, a2, s], 21, 0, 21, 24, 是, 是]], 是, 否, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum)；依完整明示資料獨立重算；feasible flow對每個cut皆有value≤capacity；同值flow/cut給雙optimality certificate；單一flow/cut不自動optimal；不計反向edges且不宣稱integrality、augmenting algorithm或一般定理已由instances證明。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r05-max-flow-min-cut-theorem-0031/)

最大流最小割定理 · qf-mod46-mfmc-flow-cut-upper-bound · candidate





## 32. 最大流最小割定理

完整有限network、全部s–t cuts與forward-edge cut capacity。操作=mfmc-flow-cut-upper-bound；資料={"vertices":["s","a0","a1","a2","a3","t"],"edges":[["s","a0"],["a0","t"],["s","a1"],["a1","t"],["s","a2"],["a2","t"],["s","a3"],["a3","t"],["a0","s"]],"source":"s","sink":"t","capacities":[8,8,9,9,10,10,11,11,13],"flows":[7,7,8,8,9,9,10,10,0]}。請依 [source_value, sink_value, flow_cut_rows, all_cut_bounds_hold, automatic_maximum_claim, certificate] 欄序回答。

cut capacity只算向外邊；flow net是向外flow減向內flow。automatic_maximum_claim的否不是宣稱這個flow一定非最大。

查看答案與解釋(34, 34, [[[s], 34, 0, 34, 38, 是, 是], [[a0, s], 34, 0, 34, 38, 是, 是], [[a1, s], 34, 0, 34, 38, 是, 是], [[a0, a1, s], 34, 0, 34, 38, 是, 是], [[a2, s], 34, 0, 34, 38, 是, 是], [[a0, a2, s], 34, 0, 34, 38, 是, 是], [[a1, a2, s], 34, 0, 34, 38, 是, 是], [[a0, a1, a2, s], 34, 0, 34, 38, 是, 是], [[a3, s], 34, 0, 34, 38, 是, 是], [[a0, a3, s], 34, 0, 34, 38, 是, 是], [[a1, a3, s], 34, 0, 34, 38, 是, 是], [[a0, a1, a3, s], 34, 0, 34, 38, 是, 是], [[a2, a3, s], 34, 0, 34, 38, 是, 是], [[a0, a2, a3, s], 34, 0, 34, 38, 是, 是], [[a1, a2, a3, s], 34, 0, 34, 38, 是, 是], [[a0, a1, a2, a3, s], 34, 0, 34, 38, 是, 是]], 是, 否, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum)

Wave6第九層核對=(34, 34, [[[s], 34, 0, 34, 38, 是, 是], [[a0, s], 34, 0, 34, 38, 是, 是], [[a1, s], 34, 0, 34, 38, 是, 是], [[a0, a1, s], 34, 0, 34, 38, 是, 是], [[a2, s], 34, 0, 34, 38, 是, 是], [[a0, a2, s], 34, 0, 34, 38, 是, 是], [[a1, a2, s], 34, 0, 34, 38, 是, 是], [[a0, a1, a2, s], 34, 0, 34, 38, 是, 是], [[a3, s], 34, 0, 34, 38, 是, 是], [[a0, a3, s], 34, 0, 34, 38, 是, 是], [[a1, a3, s], 34, 0, 34, 38, 是, 是], [[a0, a1, a3, s], 34, 0, 34, 38, 是, 是], [[a2, a3, s], 34, 0, 34, 38, 是, 是], [[a0, a2, a3, s], 34, 0, 34, 38, 是, 是], [[a1, a2, a3, s], 34, 0, 34, 38, 是, 是], [[a0, a1, a2, a3, s], 34, 0, 34, 38, 是, 是]], 是, 否, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum)；依完整明示資料獨立重算；feasible flow對每個cut皆有value≤capacity；同值flow/cut給雙optimality certificate；單一flow/cut不自動optimal；不計反向edges且不宣稱integrality、augmenting algorithm或一般定理已由instances證明。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r05-max-flow-min-cut-theorem-0032/)

最大流最小割定理 · qf-mod46-mfmc-flow-cut-upper-bound · candidate





## 33. 最大流最小割定理

完整有限network、全部s–t cuts與forward-edge cut capacity。操作=mfmc-flow-cut-upper-bound；資料={"vertices":["s","a0","a1","a2","a3","a4","t"],"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","capacities":[9,9,10,10,11,11,12,12,13,13],"flows":[8,8,9,9,10,10,11,11,12,12]}。請依 [source_value, sink_value, flow_cut_rows, all_cut_bounds_hold, automatic_maximum_claim, certificate] 欄序回答。

cut capacity只算向外邊；flow net是向外flow減向內flow。automatic_maximum_claim的否不是宣稱這個flow一定非最大。

查看答案與解釋(50, 50, [[[s], 50, 0, 50, 55, 是, 是], [[a0, s], 50, 0, 50, 55, 是, 是], [[a1, s], 50, 0, 50, 55, 是, 是], [[a0, a1, s], 50, 0, 50, 55, 是, 是], [[a2, s], 50, 0, 50, 55, 是, 是], [[a0, a2, s], 50, 0, 50, 55, 是, 是], [[a1, a2, s], 50, 0, 50, 55, 是, 是], [[a0, a1, a2, s], 50, 0, 50, 55, 是, 是], [[a3, s], 50, 0, 50, 55, 是, 是], [[a0, a3, s], 50, 0, 50, 55, 是, 是], [[a1, a3, s], 50, 0, 50, 55, 是, 是], [[a0, a1, a3, s], 50, 0, 50, 55, 是, 是], [[a2, a3, s], 50, 0, 50, 55, 是, 是], [[a0, a2, a3, s], 50, 0, 50, 55, 是, 是], [[a1, a2, a3, s], 50, 0, 50, 55, 是, 是], [[a0, a1, a2, a3, s], 50, 0, 50, 55, 是, 是], [[a4, s], 50, 0, 50, 55, 是, 是], [[a0, a4, s], 50, 0, 50, 55, 是, 是], [[a1, a4, s], 50, 0, 50, 55, 是, 是], [[a0, a1, a4, s], 50, 0, 50, 55, 是, 是], [[a2, a4, s], 50, 0, 50, 55, 是, 是], [[a0, a2, a4, s], 50, 0, 50, 55, 是, 是], [[a1, a2, a4, s], 50, 0, 50, 55, 是, 是], [[a0, a1, a2, a4, s], 50, 0, 50, 55, 是, 是], [[a3, a4, s], 50, 0, 50, 55, 是, 是], [[a0, a3, a4, s], 50, 0, 50, 55, 是, 是], [[a1, a3, a4, s], 50, 0, 50, 55, 是, 是], [[a0, a1, a3, a4, s], 50, 0, 50, 55, 是, 是], [[a2, a3, a4, s], 50, 0, 50, 55, 是, 是], [[a0, a2, a3, a4, s], 50, 0, 50, 55, 是, 是], [[a1, a2, a3, a4, s], 50, 0, 50, 55, 是, 是], [[a0, a1, a2, a3, a4, s], 50, 0, 50, 55, 是, 是]], 是, 否, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum)

Wave6第九層核對=(50, 50, [[[s], 50, 0, 50, 55, 是, 是], [[a0, s], 50, 0, 50, 55, 是, 是], [[a1, s], 50, 0, 50, 55, 是, 是], [[a0, a1, s], 50, 0, 50, 55, 是, 是], [[a2, s], 50, 0, 50, 55, 是, 是], [[a0, a2, s], 50, 0, 50, 55, 是, 是], [[a1, a2, s], 50, 0, 50, 55, 是, 是], [[a0, a1, a2, s], 50, 0, 50, 55, 是, 是], [[a3, s], 50, 0, 50, 55, 是, 是], [[a0, a3, s], 50, 0, 50, 55, 是, 是], [[a1, a3, s], 50, 0, 50, 55, 是, 是], [[a0, a1, a3, s], 50, 0, 50, 55, 是, 是], [[a2, a3, s], 50, 0, 50, 55, 是, 是], [[a0, a2, a3, s], 50, 0, 50, 55, 是, 是], [[a1, a2, a3, s], 50, 0, 50, 55, 是, 是], [[a0, a1, a2, a3, s], 50, 0, 50, 55, 是, 是], [[a4, s], 50, 0, 50, 55, 是, 是], [[a0, a4, s], 50, 0, 50, 55, 是, 是], [[a1, a4, s], 50, 0, 50, 55, 是, 是], [[a0, a1, a4, s], 50, 0, 50, 55, 是, 是], [[a2, a4, s], 50, 0, 50, 55, 是, 是], [[a0, a2, a4, s], 50, 0, 50, 55, 是, 是], [[a1, a2, a4, s], 50, 0, 50, 55, 是, 是], [[a0, a1, a2, a4, s], 50, 0, 50, 55, 是, 是], [[a3, a4, s], 50, 0, 50, 55, 是, 是], [[a0, a3, a4, s], 50, 0, 50, 55, 是, 是], [[a1, a3, a4, s], 50, 0, 50, 55, 是, 是], [[a0, a1, a3, a4, s], 50, 0, 50, 55, 是, 是], [[a2, a3, a4, s], 50, 0, 50, 55, 是, 是], [[a0, a2, a3, a4, s], 50, 0, 50, 55, 是, 是], [[a1, a2, a3, a4, s], 50, 0, 50, 55, 是, 是], [[a0, a1, a2, a3, a4, s], 50, 0, 50, 55, 是, 是]], 是, 否, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum)；依完整明示資料獨立重算；feasible flow對每個cut皆有value≤capacity；同值flow/cut給雙optimality certificate；單一flow/cut不自動optimal；不計反向edges且不宣稱integrality、augmenting algorithm或一般定理已由instances證明。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r05-max-flow-min-cut-theorem-0033/)

最大流最小割定理 · qf-mod46-mfmc-flow-cut-upper-bound · candidate





## 34. 最大流最小割定理

完整有限network、全部s–t cuts與forward-edge cut capacity。操作=mfmc-flow-cut-upper-bound；資料={"vertices":["s","a0","a1","t"],"edges":[["s","a0"],["a0","t"],["s","a1"],["a1","t"]],"source":"s","sink":"t","capacities":[10,10,11,11],"flows":[9,9,10,10]}。請依 [source_value, sink_value, flow_cut_rows, all_cut_bounds_hold, automatic_maximum_claim, certificate] 欄序回答。

cut capacity只算向外邊；flow net是向外flow減向內flow。automatic_maximum_claim的否不是宣稱這個flow一定非最大。

查看答案與解釋(19, 19, [[[s], 19, 0, 19, 21, 是, 是], [[a0, s], 19, 0, 19, 21, 是, 是], [[a1, s], 19, 0, 19, 21, 是, 是], [[a0, a1, s], 19, 0, 19, 21, 是, 是]], 是, 否, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum)

Wave6第九層核對=(19, 19, [[[s], 19, 0, 19, 21, 是, 是], [[a0, s], 19, 0, 19, 21, 是, 是], [[a1, s], 19, 0, 19, 21, 是, 是], [[a0, a1, s], 19, 0, 19, 21, 是, 是]], 是, 否, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum)；依完整明示資料獨立重算；feasible flow對每個cut皆有value≤capacity；同值flow/cut給雙optimality certificate；單一flow/cut不自動optimal；不計反向edges且不宣稱integrality、augmenting algorithm或一般定理已由instances證明。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r05-max-flow-min-cut-theorem-0034/)

最大流最小割定理 · qf-mod46-mfmc-flow-cut-upper-bound · candidate





## 35. 最大流最小割定理

完整有限network、全部s–t cuts與forward-edge cut capacity。操作=mfmc-flow-cut-upper-bound；資料={"vertices":["s","a0","a1","a2","t"],"edges":[["s","a0"],["a0","t"],["s","a1"],["a1","t"],["s","a2"],["a2","t"],["a0","s"]],"source":"s","sink":"t","capacities":[11,11,12,12,13,13,16],"flows":[10,10,11,11,12,12,0]}。請依 [source_value, sink_value, flow_cut_rows, all_cut_bounds_hold, automatic_maximum_claim, certificate] 欄序回答。

cut capacity只算向外邊；flow net是向外flow減向內flow。automatic_maximum_claim的否不是宣稱這個flow一定非最大。

查看答案與解釋(33, 33, [[[s], 33, 0, 33, 36, 是, 是], [[a0, s], 33, 0, 33, 36, 是, 是], [[a1, s], 33, 0, 33, 36, 是, 是], [[a0, a1, s], 33, 0, 33, 36, 是, 是], [[a2, s], 33, 0, 33, 36, 是, 是], [[a0, a2, s], 33, 0, 33, 36, 是, 是], [[a1, a2, s], 33, 0, 33, 36, 是, 是], [[a0, a1, a2, s], 33, 0, 33, 36, 是, 是]], 是, 否, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum)

Wave6第九層核對=(33, 33, [[[s], 33, 0, 33, 36, 是, 是], [[a0, s], 33, 0, 33, 36, 是, 是], [[a1, s], 33, 0, 33, 36, 是, 是], [[a0, a1, s], 33, 0, 33, 36, 是, 是], [[a2, s], 33, 0, 33, 36, 是, 是], [[a0, a2, s], 33, 0, 33, 36, 是, 是], [[a1, a2, s], 33, 0, 33, 36, 是, 是], [[a0, a1, a2, s], 33, 0, 33, 36, 是, 是]], 是, 否, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum)；依完整明示資料獨立重算；feasible flow對每個cut皆有value≤capacity；同值flow/cut給雙optimality certificate；單一flow/cut不自動optimal；不計反向edges且不宣稱integrality、augmenting algorithm或一般定理已由instances證明。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r05-max-flow-min-cut-theorem-0035/)

最大流最小割定理 · qf-mod46-mfmc-flow-cut-upper-bound · candidate





## 36. 最大流最小割定理

完整有限network、全部s–t cuts與forward-edge cut capacity。操作=mfmc-flow-cut-upper-bound；資料={"vertices":["s","a0","a1","a2","a3","t"],"edges":[["s","a0"],["a0","t"],["s","a1"],["a1","t"],["s","a2"],["a2","t"],["s","a3"],["a3","t"]],"source":"s","sink":"t","capacities":[12,12,13,13,14,14,15,15],"flows":[11,11,12,12,13,13,14,14]}。請依 [source_value, sink_value, flow_cut_rows, all_cut_bounds_hold, automatic_maximum_claim, certificate] 欄序回答。

cut capacity只算向外邊；flow net是向外flow減向內flow。automatic_maximum_claim的否不是宣稱這個flow一定非最大。

查看答案與解釋(50, 50, [[[s], 50, 0, 50, 54, 是, 是], [[a0, s], 50, 0, 50, 54, 是, 是], [[a1, s], 50, 0, 50, 54, 是, 是], [[a0, a1, s], 50, 0, 50, 54, 是, 是], [[a2, s], 50, 0, 50, 54, 是, 是], [[a0, a2, s], 50, 0, 50, 54, 是, 是], [[a1, a2, s], 50, 0, 50, 54, 是, 是], [[a0, a1, a2, s], 50, 0, 50, 54, 是, 是], [[a3, s], 50, 0, 50, 54, 是, 是], [[a0, a3, s], 50, 0, 50, 54, 是, 是], [[a1, a3, s], 50, 0, 50, 54, 是, 是], [[a0, a1, a3, s], 50, 0, 50, 54, 是, 是], [[a2, a3, s], 50, 0, 50, 54, 是, 是], [[a0, a2, a3, s], 50, 0, 50, 54, 是, 是], [[a1, a2, a3, s], 50, 0, 50, 54, 是, 是], [[a0, a1, a2, a3, s], 50, 0, 50, 54, 是, 是]], 是, 否, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum)

Wave6第九層核對=(50, 50, [[[s], 50, 0, 50, 54, 是, 是], [[a0, s], 50, 0, 50, 54, 是, 是], [[a1, s], 50, 0, 50, 54, 是, 是], [[a0, a1, s], 50, 0, 50, 54, 是, 是], [[a2, s], 50, 0, 50, 54, 是, 是], [[a0, a2, s], 50, 0, 50, 54, 是, 是], [[a1, a2, s], 50, 0, 50, 54, 是, 是], [[a0, a1, a2, s], 50, 0, 50, 54, 是, 是], [[a3, s], 50, 0, 50, 54, 是, 是], [[a0, a3, s], 50, 0, 50, 54, 是, 是], [[a1, a3, s], 50, 0, 50, 54, 是, 是], [[a0, a1, a3, s], 50, 0, 50, 54, 是, 是], [[a2, a3, s], 50, 0, 50, 54, 是, 是], [[a0, a2, a3, s], 50, 0, 50, 54, 是, 是], [[a1, a2, a3, s], 50, 0, 50, 54, 是, 是], [[a0, a1, a2, a3, s], 50, 0, 50, 54, 是, 是]], 是, 否, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum)；依完整明示資料獨立重算；feasible flow對每個cut皆有value≤capacity；同值flow/cut給雙optimality certificate；單一flow/cut不自動optimal；不計反向edges且不宣稱integrality、augmenting algorithm或一般定理已由instances證明。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r05-max-flow-min-cut-theorem-0036/)

最大流最小割定理 · qf-mod46-mfmc-flow-cut-upper-bound · candidate





## 37. 最大流最小割定理

完整有限network、全部s–t cuts與forward-edge cut capacity。操作=mfmc-flow-cut-upper-bound；資料={"vertices":["s","a0","a1","a2","a3","a4","t"],"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","capacities":[13,13,14,14,15,15,16,16,17,17],"flows":[12,12,13,13,14,14,15,15,16,16]}。請依 [source_value, sink_value, flow_cut_rows, all_cut_bounds_hold, automatic_maximum_claim, certificate] 欄序回答。

cut capacity只算向外邊；flow net是向外flow減向內flow。automatic_maximum_claim的否不是宣稱這個flow一定非最大。

查看答案與解釋(70, 70, [[[s], 70, 0, 70, 75, 是, 是], [[a0, s], 70, 0, 70, 75, 是, 是], [[a1, s], 70, 0, 70, 75, 是, 是], [[a0, a1, s], 70, 0, 70, 75, 是, 是], [[a2, s], 70, 0, 70, 75, 是, 是], [[a0, a2, s], 70, 0, 70, 75, 是, 是], [[a1, a2, s], 70, 0, 70, 75, 是, 是], [[a0, a1, a2, s], 70, 0, 70, 75, 是, 是], [[a3, s], 70, 0, 70, 75, 是, 是], [[a0, a3, s], 70, 0, 70, 75, 是, 是], [[a1, a3, s], 70, 0, 70, 75, 是, 是], [[a0, a1, a3, s], 70, 0, 70, 75, 是, 是], [[a2, a3, s], 70, 0, 70, 75, 是, 是], [[a0, a2, a3, s], 70, 0, 70, 75, 是, 是], [[a1, a2, a3, s], 70, 0, 70, 75, 是, 是], [[a0, a1, a2, a3, s], 70, 0, 70, 75, 是, 是], [[a4, s], 70, 0, 70, 75, 是, 是], [[a0, a4, s], 70, 0, 70, 75, 是, 是], [[a1, a4, s], 70, 0, 70, 75, 是, 是], [[a0, a1, a4, s], 70, 0, 70, 75, 是, 是], [[a2, a4, s], 70, 0, 70, 75, 是, 是], [[a0, a2, a4, s], 70, 0, 70, 75, 是, 是], [[a1, a2, a4, s], 70, 0, 70, 75, 是, 是], [[a0, a1, a2, a4, s], 70, 0, 70, 75, 是, 是], [[a3, a4, s], 70, 0, 70, 75, 是, 是], [[a0, a3, a4, s], 70, 0, 70, 75, 是, 是], [[a1, a3, a4, s], 70, 0, 70, 75, 是, 是], [[a0, a1, a3, a4, s], 70, 0, 70, 75, 是, 是], [[a2, a3, a4, s], 70, 0, 70, 75, 是, 是], [[a0, a2, a3, a4, s], 70, 0, 70, 75, 是, 是], [[a1, a2, a3, a4, s], 70, 0, 70, 75, 是, 是], [[a0, a1, a2, a3, a4, s], 70, 0, 70, 75, 是, 是]], 是, 否, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum)

Wave6第九層核對=(70, 70, [[[s], 70, 0, 70, 75, 是, 是], [[a0, s], 70, 0, 70, 75, 是, 是], [[a1, s], 70, 0, 70, 75, 是, 是], [[a0, a1, s], 70, 0, 70, 75, 是, 是], [[a2, s], 70, 0, 70, 75, 是, 是], [[a0, a2, s], 70, 0, 70, 75, 是, 是], [[a1, a2, s], 70, 0, 70, 75, 是, 是], [[a0, a1, a2, s], 70, 0, 70, 75, 是, 是], [[a3, s], 70, 0, 70, 75, 是, 是], [[a0, a3, s], 70, 0, 70, 75, 是, 是], [[a1, a3, s], 70, 0, 70, 75, 是, 是], [[a0, a1, a3, s], 70, 0, 70, 75, 是, 是], [[a2, a3, s], 70, 0, 70, 75, 是, 是], [[a0, a2, a3, s], 70, 0, 70, 75, 是, 是], [[a1, a2, a3, s], 70, 0, 70, 75, 是, 是], [[a0, a1, a2, a3, s], 70, 0, 70, 75, 是, 是], [[a4, s], 70, 0, 70, 75, 是, 是], [[a0, a4, s], 70, 0, 70, 75, 是, 是], [[a1, a4, s], 70, 0, 70, 75, 是, 是], [[a0, a1, a4, s], 70, 0, 70, 75, 是, 是], [[a2, a4, s], 70, 0, 70, 75, 是, 是], [[a0, a2, a4, s], 70, 0, 70, 75, 是, 是], [[a1, a2, a4, s], 70, 0, 70, 75, 是, 是], [[a0, a1, a2, a4, s], 70, 0, 70, 75, 是, 是], [[a3, a4, s], 70, 0, 70, 75, 是, 是], [[a0, a3, a4, s], 70, 0, 70, 75, 是, 是], [[a1, a3, a4, s], 70, 0, 70, 75, 是, 是], [[a0, a1, a3, a4, s], 70, 0, 70, 75, 是, 是], [[a2, a3, a4, s], 70, 0, 70, 75, 是, 是], [[a0, a2, a3, a4, s], 70, 0, 70, 75, 是, 是], [[a1, a2, a3, a4, s], 70, 0, 70, 75, 是, 是], [[a0, a1, a2, a3, a4, s], 70, 0, 70, 75, 是, 是]], 是, 否, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum)；依完整明示資料獨立重算；feasible flow對每個cut皆有value≤capacity；同值flow/cut給雙optimality certificate；單一flow/cut不自動optimal；不計反向edges且不宣稱integrality、augmenting algorithm或一般定理已由instances證明。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r05-max-flow-min-cut-theorem-0037/)

最大流最小割定理 · qf-mod46-mfmc-flow-cut-upper-bound · candidate





## 38. 最大流最小割定理

完整有限network、全部s–t cuts與forward-edge cut capacity。操作=mfmc-flow-cut-upper-bound；資料={"vertices":["s","a0","a1","t"],"edges":[["s","a0"],["a0","t"],["s","a1"],["a1","t"],["a0","s"]],"source":"s","sink":"t","capacities":[14,14,15,15,19],"flows":[13,13,14,14,0]}。請依 [source_value, sink_value, flow_cut_rows, all_cut_bounds_hold, automatic_maximum_claim, certificate] 欄序回答。

cut capacity只算向外邊；flow net是向外flow減向內flow。automatic_maximum_claim的否不是宣稱這個flow一定非最大。

查看答案與解釋(27, 27, [[[s], 27, 0, 27, 29, 是, 是], [[a0, s], 27, 0, 27, 29, 是, 是], [[a1, s], 27, 0, 27, 29, 是, 是], [[a0, a1, s], 27, 0, 27, 29, 是, 是]], 是, 否, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum)

Wave6第九層核對=(27, 27, [[[s], 27, 0, 27, 29, 是, 是], [[a0, s], 27, 0, 27, 29, 是, 是], [[a1, s], 27, 0, 27, 29, 是, 是], [[a0, a1, s], 27, 0, 27, 29, 是, 是]], 是, 否, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum)；依完整明示資料獨立重算；feasible flow對每個cut皆有value≤capacity；同值flow/cut給雙optimality certificate；單一flow/cut不自動optimal；不計反向edges且不宣稱integrality、augmenting algorithm或一般定理已由instances證明。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r05-max-flow-min-cut-theorem-0038/)

最大流最小割定理 · qf-mod46-mfmc-flow-cut-upper-bound · candidate





## 39. 最大流最小割定理

完整有限network、全部s–t cuts與forward-edge cut capacity。操作=mfmc-flow-cut-upper-bound；資料={"vertices":["s","a0","a1","a2","t"],"edges":[["s","a0"],["a0","t"],["s","a1"],["a1","t"],["s","a2"],["a2","t"]],"source":"s","sink":"t","capacities":[15,15,16,16,17,17],"flows":[14,14,15,15,16,16]}。請依 [source_value, sink_value, flow_cut_rows, all_cut_bounds_hold, automatic_maximum_claim, certificate] 欄序回答。

cut capacity只算向外邊；flow net是向外flow減向內flow。automatic_maximum_claim的否不是宣稱這個flow一定非最大。

查看答案與解釋(45, 45, [[[s], 45, 0, 45, 48, 是, 是], [[a0, s], 45, 0, 45, 48, 是, 是], [[a1, s], 45, 0, 45, 48, 是, 是], [[a0, a1, s], 45, 0, 45, 48, 是, 是], [[a2, s], 45, 0, 45, 48, 是, 是], [[a0, a2, s], 45, 0, 45, 48, 是, 是], [[a1, a2, s], 45, 0, 45, 48, 是, 是], [[a0, a1, a2, s], 45, 0, 45, 48, 是, 是]], 是, 否, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum)

Wave6第九層核對=(45, 45, [[[s], 45, 0, 45, 48, 是, 是], [[a0, s], 45, 0, 45, 48, 是, 是], [[a1, s], 45, 0, 45, 48, 是, 是], [[a0, a1, s], 45, 0, 45, 48, 是, 是], [[a2, s], 45, 0, 45, 48, 是, 是], [[a0, a2, s], 45, 0, 45, 48, 是, 是], [[a1, a2, s], 45, 0, 45, 48, 是, 是], [[a0, a1, a2, s], 45, 0, 45, 48, 是, 是]], 是, 否, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum)；依完整明示資料獨立重算；feasible flow對每個cut皆有value≤capacity；同值flow/cut給雙optimality certificate；單一flow/cut不自動optimal；不計反向edges且不宣稱integrality、augmenting algorithm或一般定理已由instances證明。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r05-max-flow-min-cut-theorem-0039/)

最大流最小割定理 · qf-mod46-mfmc-flow-cut-upper-bound · candidate





## 40. 最大流最小割定理

完整有限network、全部s–t cuts與forward-edge cut capacity。操作=mfmc-flow-cut-upper-bound；資料={"vertices":["s","a0","a1","a2","a3","t"],"edges":[["s","a0"],["a0","t"],["s","a1"],["a1","t"],["s","a2"],["a2","t"],["s","a3"],["a3","t"]],"source":"s","sink":"t","capacities":[16,16,17,17,18,18,19,19],"flows":[15,15,16,16,17,17,18,18]}。請依 [source_value, sink_value, flow_cut_rows, all_cut_bounds_hold, automatic_maximum_claim, certificate] 欄序回答。

cut capacity只算向外邊；flow net是向外flow減向內flow。automatic_maximum_claim的否不是宣稱這個flow一定非最大。

查看答案與解釋(66, 66, [[[s], 66, 0, 66, 70, 是, 是], [[a0, s], 66, 0, 66, 70, 是, 是], [[a1, s], 66, 0, 66, 70, 是, 是], [[a0, a1, s], 66, 0, 66, 70, 是, 是], [[a2, s], 66, 0, 66, 70, 是, 是], [[a0, a2, s], 66, 0, 66, 70, 是, 是], [[a1, a2, s], 66, 0, 66, 70, 是, 是], [[a0, a1, a2, s], 66, 0, 66, 70, 是, 是], [[a3, s], 66, 0, 66, 70, 是, 是], [[a0, a3, s], 66, 0, 66, 70, 是, 是], [[a1, a3, s], 66, 0, 66, 70, 是, 是], [[a0, a1, a3, s], 66, 0, 66, 70, 是, 是], [[a2, a3, s], 66, 0, 66, 70, 是, 是], [[a0, a2, a3, s], 66, 0, 66, 70, 是, 是], [[a1, a2, a3, s], 66, 0, 66, 70, 是, 是], [[a0, a1, a2, a3, s], 66, 0, 66, 70, 是, 是]], 是, 否, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum)

Wave6第九層核對=(66, 66, [[[s], 66, 0, 66, 70, 是, 是], [[a0, s], 66, 0, 66, 70, 是, 是], [[a1, s], 66, 0, 66, 70, 是, 是], [[a0, a1, s], 66, 0, 66, 70, 是, 是], [[a2, s], 66, 0, 66, 70, 是, 是], [[a0, a2, s], 66, 0, 66, 70, 是, 是], [[a1, a2, s], 66, 0, 66, 70, 是, 是], [[a0, a1, a2, s], 66, 0, 66, 70, 是, 是], [[a3, s], 66, 0, 66, 70, 是, 是], [[a0, a3, s], 66, 0, 66, 70, 是, 是], [[a1, a3, s], 66, 0, 66, 70, 是, 是], [[a0, a1, a3, s], 66, 0, 66, 70, 是, 是], [[a2, a3, s], 66, 0, 66, 70, 是, 是], [[a0, a2, a3, s], 66, 0, 66, 70, 是, 是], [[a1, a2, a3, s], 66, 0, 66, 70, 是, 是], [[a0, a1, a2, a3, s], 66, 0, 66, 70, 是, 是]], 是, 否, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum)；依完整明示資料獨立重算；feasible flow對每個cut皆有value≤capacity；同值flow/cut給雙optimality certificate；單一flow/cut不自動optimal；不計反向edges且不宣稱integrality、augmenting algorithm或一般定理已由instances證明。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r05-max-flow-min-cut-theorem-0040/)

最大流最小割定理 · qf-mod46-mfmc-flow-cut-upper-bound · candidate





## 41. 最大流最小割定理

完整有限network、全部s–t cuts與forward-edge cut capacity。操作=mfmc-flow-cut-upper-bound；資料={"vertices":["s","a0","a1","a2","a3","a4","t"],"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","capacities":[17,17,18,18,19,19,20,20,21,21,22],"flows":[16,16,17,17,18,18,19,19,20,20,0]}。請依 [source_value, sink_value, flow_cut_rows, all_cut_bounds_hold, automatic_maximum_claim, certificate] 欄序回答。

cut capacity只算向外邊；flow net是向外flow減向內flow。automatic_maximum_claim的否不是宣稱這個flow一定非最大。

查看答案與解釋(90, 90, [[[s], 90, 0, 90, 95, 是, 是], [[a0, s], 90, 0, 90, 95, 是, 是], [[a1, s], 90, 0, 90, 95, 是, 是], [[a0, a1, s], 90, 0, 90, 95, 是, 是], [[a2, s], 90, 0, 90, 95, 是, 是], [[a0, a2, s], 90, 0, 90, 95, 是, 是], [[a1, a2, s], 90, 0, 90, 95, 是, 是], [[a0, a1, a2, s], 90, 0, 90, 95, 是, 是], [[a3, s], 90, 0, 90, 95, 是, 是], [[a0, a3, s], 90, 0, 90, 95, 是, 是], [[a1, a3, s], 90, 0, 90, 95, 是, 是], [[a0, a1, a3, s], 90, 0, 90, 95, 是, 是], [[a2, a3, s], 90, 0, 90, 95, 是, 是], [[a0, a2, a3, s], 90, 0, 90, 95, 是, 是], [[a1, a2, a3, s], 90, 0, 90, 95, 是, 是], [[a0, a1, a2, a3, s], 90, 0, 90, 95, 是, 是], [[a4, s], 90, 0, 90, 95, 是, 是], [[a0, a4, s], 90, 0, 90, 95, 是, 是], [[a1, a4, s], 90, 0, 90, 95, 是, 是], [[a0, a1, a4, s], 90, 0, 90, 95, 是, 是], [[a2, a4, s], 90, 0, 90, 95, 是, 是], [[a0, a2, a4, s], 90, 0, 90, 95, 是, 是], [[a1, a2, a4, s], 90, 0, 90, 95, 是, 是], [[a0, a1, a2, a4, s], 90, 0, 90, 95, 是, 是], [[a3, a4, s], 90, 0, 90, 95, 是, 是], [[a0, a3, a4, s], 90, 0, 90, 95, 是, 是], [[a1, a3, a4, s], 90, 0, 90, 95, 是, 是], [[a0, a1, a3, a4, s], 90, 0, 90, 95, 是, 是], [[a2, a3, a4, s], 90, 0, 90, 95, 是, 是], [[a0, a2, a3, a4, s], 90, 0, 90, 95, 是, 是], [[a1, a2, a3, a4, s], 90, 0, 90, 95, 是, 是], [[a0, a1, a2, a3, a4, s], 90, 0, 90, 95, 是, 是]], 是, 否, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum)

Wave6第九層核對=(90, 90, [[[s], 90, 0, 90, 95, 是, 是], [[a0, s], 90, 0, 90, 95, 是, 是], [[a1, s], 90, 0, 90, 95, 是, 是], [[a0, a1, s], 90, 0, 90, 95, 是, 是], [[a2, s], 90, 0, 90, 95, 是, 是], [[a0, a2, s], 90, 0, 90, 95, 是, 是], [[a1, a2, s], 90, 0, 90, 95, 是, 是], [[a0, a1, a2, s], 90, 0, 90, 95, 是, 是], [[a3, s], 90, 0, 90, 95, 是, 是], [[a0, a3, s], 90, 0, 90, 95, 是, 是], [[a1, a3, s], 90, 0, 90, 95, 是, 是], [[a0, a1, a3, s], 90, 0, 90, 95, 是, 是], [[a2, a3, s], 90, 0, 90, 95, 是, 是], [[a0, a2, a3, s], 90, 0, 90, 95, 是, 是], [[a1, a2, a3, s], 90, 0, 90, 95, 是, 是], [[a0, a1, a2, a3, s], 90, 0, 90, 95, 是, 是], [[a4, s], 90, 0, 90, 95, 是, 是], [[a0, a4, s], 90, 0, 90, 95, 是, 是], [[a1, a4, s], 90, 0, 90, 95, 是, 是], [[a0, a1, a4, s], 90, 0, 90, 95, 是, 是], [[a2, a4, s], 90, 0, 90, 95, 是, 是], [[a0, a2, a4, s], 90, 0, 90, 95, 是, 是], [[a1, a2, a4, s], 90, 0, 90, 95, 是, 是], [[a0, a1, a2, a4, s], 90, 0, 90, 95, 是, 是], [[a3, a4, s], 90, 0, 90, 95, 是, 是], [[a0, a3, a4, s], 90, 0, 90, 95, 是, 是], [[a1, a3, a4, s], 90, 0, 90, 95, 是, 是], [[a0, a1, a3, a4, s], 90, 0, 90, 95, 是, 是], [[a2, a3, a4, s], 90, 0, 90, 95, 是, 是], [[a0, a2, a3, a4, s], 90, 0, 90, 95, 是, 是], [[a1, a2, a3, a4, s], 90, 0, 90, 95, 是, 是], [[a0, a1, a2, a3, a4, s], 90, 0, 90, 95, 是, 是]], 是, 否, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum)；依完整明示資料獨立重算；feasible flow對每個cut皆有value≤capacity；同值flow/cut給雙optimality certificate；單一flow/cut不自動optimal；不計反向edges且不宣稱integrality、augmenting algorithm或一般定理已由instances證明。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r05-max-flow-min-cut-theorem-0041/)

最大流最小割定理 · qf-mod46-mfmc-flow-cut-upper-bound · candidate





## 42. 最大流最小割定理

完整有限network、全部s–t cuts與forward-edge cut capacity。操作=mfmc-flow-cut-upper-bound；資料={"vertices":["s","a0","a1","t"],"edges":[["s","a0"],["a0","t"],["s","a1"],["a1","t"]],"source":"s","sink":"t","capacities":[18,18,19,19],"flows":[17,17,18,18]}。請依 [source_value, sink_value, flow_cut_rows, all_cut_bounds_hold, automatic_maximum_claim, certificate] 欄序回答。

cut capacity只算向外邊；flow net是向外flow減向內flow。automatic_maximum_claim的否不是宣稱這個flow一定非最大。

查看答案與解釋(35, 35, [[[s], 35, 0, 35, 37, 是, 是], [[a0, s], 35, 0, 35, 37, 是, 是], [[a1, s], 35, 0, 35, 37, 是, 是], [[a0, a1, s], 35, 0, 35, 37, 是, 是]], 是, 否, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum)

Wave6第九層核對=(35, 35, [[[s], 35, 0, 35, 37, 是, 是], [[a0, s], 35, 0, 35, 37, 是, 是], [[a1, s], 35, 0, 35, 37, 是, 是], [[a0, a1, s], 35, 0, 35, 37, 是, 是]], 是, 否, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum)；依完整明示資料獨立重算；feasible flow對每個cut皆有value≤capacity；同值flow/cut給雙optimality certificate；單一flow/cut不自動optimal；不計反向edges且不宣稱integrality、augmenting algorithm或一般定理已由instances證明。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r05-max-flow-min-cut-theorem-0042/)

最大流最小割定理 · qf-mod46-mfmc-flow-cut-upper-bound · candidate





## 43. 最大流最小割定理

完整有限network、全部s–t cuts與forward-edge cut capacity。操作=mfmc-flow-cut-upper-bound；資料={"vertices":["s","a0","a1","a2","t"],"edges":[["s","a0"],["a0","t"],["s","a1"],["a1","t"],["s","a2"],["a2","t"]],"source":"s","sink":"t","capacities":[19,19,20,20,21,21],"flows":[18,18,19,19,20,20]}。請依 [source_value, sink_value, flow_cut_rows, all_cut_bounds_hold, automatic_maximum_claim, certificate] 欄序回答。

cut capacity只算向外邊；flow net是向外flow減向內flow。automatic_maximum_claim的否不是宣稱這個flow一定非最大。

查看答案與解釋(57, 57, [[[s], 57, 0, 57, 60, 是, 是], [[a0, s], 57, 0, 57, 60, 是, 是], [[a1, s], 57, 0, 57, 60, 是, 是], [[a0, a1, s], 57, 0, 57, 60, 是, 是], [[a2, s], 57, 0, 57, 60, 是, 是], [[a0, a2, s], 57, 0, 57, 60, 是, 是], [[a1, a2, s], 57, 0, 57, 60, 是, 是], [[a0, a1, a2, s], 57, 0, 57, 60, 是, 是]], 是, 否, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum)

Wave6第九層核對=(57, 57, [[[s], 57, 0, 57, 60, 是, 是], [[a0, s], 57, 0, 57, 60, 是, 是], [[a1, s], 57, 0, 57, 60, 是, 是], [[a0, a1, s], 57, 0, 57, 60, 是, 是], [[a2, s], 57, 0, 57, 60, 是, 是], [[a0, a2, s], 57, 0, 57, 60, 是, 是], [[a1, a2, s], 57, 0, 57, 60, 是, 是], [[a0, a1, a2, s], 57, 0, 57, 60, 是, 是]], 是, 否, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum)；依完整明示資料獨立重算；feasible flow對每個cut皆有value≤capacity；同值flow/cut給雙optimality certificate；單一flow/cut不自動optimal；不計反向edges且不宣稱integrality、augmenting algorithm或一般定理已由instances證明。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r05-max-flow-min-cut-theorem-0043/)

最大流最小割定理 · qf-mod46-mfmc-flow-cut-upper-bound · candidate





## 44. 最大流最小割定理

完整有限network、全部s–t cuts與forward-edge cut capacity。操作=mfmc-flow-cut-upper-bound；資料={"vertices":["s","a0","a1","a2","a3","t"],"edges":[["s","a0"],["a0","t"],["s","a1"],["a1","t"],["s","a2"],["a2","t"],["s","a3"],["a3","t"],["a0","s"]],"source":"s","sink":"t","capacities":[20,20,21,21,22,22,23,23,25],"flows":[19,19,20,20,21,21,22,22,0]}。請依 [source_value, sink_value, flow_cut_rows, all_cut_bounds_hold, automatic_maximum_claim, certificate] 欄序回答。

cut capacity只算向外邊；flow net是向外flow減向內flow。automatic_maximum_claim的否不是宣稱這個flow一定非最大。

查看答案與解釋(82, 82, [[[s], 82, 0, 82, 86, 是, 是], [[a0, s], 82, 0, 82, 86, 是, 是], [[a1, s], 82, 0, 82, 86, 是, 是], [[a0, a1, s], 82, 0, 82, 86, 是, 是], [[a2, s], 82, 0, 82, 86, 是, 是], [[a0, a2, s], 82, 0, 82, 86, 是, 是], [[a1, a2, s], 82, 0, 82, 86, 是, 是], [[a0, a1, a2, s], 82, 0, 82, 86, 是, 是], [[a3, s], 82, 0, 82, 86, 是, 是], [[a0, a3, s], 82, 0, 82, 86, 是, 是], [[a1, a3, s], 82, 0, 82, 86, 是, 是], [[a0, a1, a3, s], 82, 0, 82, 86, 是, 是], [[a2, a3, s], 82, 0, 82, 86, 是, 是], [[a0, a2, a3, s], 82, 0, 82, 86, 是, 是], [[a1, a2, a3, s], 82, 0, 82, 86, 是, 是], [[a0, a1, a2, a3, s], 82, 0, 82, 86, 是, 是]], 是, 否, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum)

Wave6第九層核對=(82, 82, [[[s], 82, 0, 82, 86, 是, 是], [[a0, s], 82, 0, 82, 86, 是, 是], [[a1, s], 82, 0, 82, 86, 是, 是], [[a0, a1, s], 82, 0, 82, 86, 是, 是], [[a2, s], 82, 0, 82, 86, 是, 是], [[a0, a2, s], 82, 0, 82, 86, 是, 是], [[a1, a2, s], 82, 0, 82, 86, 是, 是], [[a0, a1, a2, s], 82, 0, 82, 86, 是, 是], [[a3, s], 82, 0, 82, 86, 是, 是], [[a0, a3, s], 82, 0, 82, 86, 是, 是], [[a1, a3, s], 82, 0, 82, 86, 是, 是], [[a0, a1, a3, s], 82, 0, 82, 86, 是, 是], [[a2, a3, s], 82, 0, 82, 86, 是, 是], [[a0, a2, a3, s], 82, 0, 82, 86, 是, 是], [[a1, a2, a3, s], 82, 0, 82, 86, 是, 是], [[a0, a1, a2, a3, s], 82, 0, 82, 86, 是, 是]], 是, 否, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum)；依完整明示資料獨立重算；feasible flow對每個cut皆有value≤capacity；同值flow/cut給雙optimality certificate；單一flow/cut不自動optimal；不計反向edges且不宣稱integrality、augmenting algorithm或一般定理已由instances證明。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r05-max-flow-min-cut-theorem-0044/)

最大流最小割定理 · qf-mod46-mfmc-flow-cut-upper-bound · candidate





## 45. 最大流最小割定理

完整有限network、全部s–t cuts與forward-edge cut capacity。操作=mfmc-flow-cut-upper-bound；資料={"vertices":["s","a0","a1","a2","a3","a4","t"],"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","capacities":[21,21,22,22,23,23,24,24,25,25],"flows":[20,20,21,21,22,22,23,23,24,24]}。請依 [source_value, sink_value, flow_cut_rows, all_cut_bounds_hold, automatic_maximum_claim, certificate] 欄序回答。

cut capacity只算向外邊；flow net是向外flow減向內flow。automatic_maximum_claim的否不是宣稱這個flow一定非最大。

查看答案與解釋(110, 110, [[[s], 110, 0, 110, 115, 是, 是], [[a0, s], 110, 0, 110, 115, 是, 是], [[a1, s], 110, 0, 110, 115, 是, 是], [[a0, a1, s], 110, 0, 110, 115, 是, 是], [[a2, s], 110, 0, 110, 115, 是, 是], [[a0, a2, s], 110, 0, 110, 115, 是, 是], [[a1, a2, s], 110, 0, 110, 115, 是, 是], [[a0, a1, a2, s], 110, 0, 110, 115, 是, 是], [[a3, s], 110, 0, 110, 115, 是, 是], [[a0, a3, s], 110, 0, 110, 115, 是, 是], [[a1, a3, s], 110, 0, 110, 115, 是, 是], [[a0, a1, a3, s], 110, 0, 110, 115, 是, 是], [[a2, a3, s], 110, 0, 110, 115, 是, 是], [[a0, a2, a3, s], 110, 0, 110, 115, 是, 是], [[a1, a2, a3, s], 110, 0, 110, 115, 是, 是], [[a0, a1, a2, a3, s], 110, 0, 110, 115, 是, 是], [[a4, s], 110, 0, 110, 115, 是, 是], [[a0, a4, s], 110, 0, 110, 115, 是, 是], [[a1, a4, s], 110, 0, 110, 115, 是, 是], [[a0, a1, a4, s], 110, 0, 110, 115, 是, 是], [[a2, a4, s], 110, 0, 110, 115, 是, 是], [[a0, a2, a4, s], 110, 0, 110, 115, 是, 是], [[a1, a2, a4, s], 110, 0, 110, 115, 是, 是], [[a0, a1, a2, a4, s], 110, 0, 110, 115, 是, 是], [[a3, a4, s], 110, 0, 110, 115, 是, 是], [[a0, a3, a4, s], 110, 0, 110, 115, 是, 是], [[a1, a3, a4, s], 110, 0, 110, 115, 是, 是], [[a0, a1, a3, a4, s], 110, 0, 110, 115, 是, 是], [[a2, a3, a4, s], 110, 0, 110, 115, 是, 是], [[a0, a2, a3, a4, s], 110, 0, 110, 115, 是, 是], [[a1, a2, a3, a4, s], 110, 0, 110, 115, 是, 是], [[a0, a1, a2, a3, a4, s], 110, 0, 110, 115, 是, 是]], 是, 否, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum)

Wave6第九層核對=(110, 110, [[[s], 110, 0, 110, 115, 是, 是], [[a0, s], 110, 0, 110, 115, 是, 是], [[a1, s], 110, 0, 110, 115, 是, 是], [[a0, a1, s], 110, 0, 110, 115, 是, 是], [[a2, s], 110, 0, 110, 115, 是, 是], [[a0, a2, s], 110, 0, 110, 115, 是, 是], [[a1, a2, s], 110, 0, 110, 115, 是, 是], [[a0, a1, a2, s], 110, 0, 110, 115, 是, 是], [[a3, s], 110, 0, 110, 115, 是, 是], [[a0, a3, s], 110, 0, 110, 115, 是, 是], [[a1, a3, s], 110, 0, 110, 115, 是, 是], [[a0, a1, a3, s], 110, 0, 110, 115, 是, 是], [[a2, a3, s], 110, 0, 110, 115, 是, 是], [[a0, a2, a3, s], 110, 0, 110, 115, 是, 是], [[a1, a2, a3, s], 110, 0, 110, 115, 是, 是], [[a0, a1, a2, a3, s], 110, 0, 110, 115, 是, 是], [[a4, s], 110, 0, 110, 115, 是, 是], [[a0, a4, s], 110, 0, 110, 115, 是, 是], [[a1, a4, s], 110, 0, 110, 115, 是, 是], [[a0, a1, a4, s], 110, 0, 110, 115, 是, 是], [[a2, a4, s], 110, 0, 110, 115, 是, 是], [[a0, a2, a4, s], 110, 0, 110, 115, 是, 是], [[a1, a2, a4, s], 110, 0, 110, 115, 是, 是], [[a0, a1, a2, a4, s], 110, 0, 110, 115, 是, 是], [[a3, a4, s], 110, 0, 110, 115, 是, 是], [[a0, a3, a4, s], 110, 0, 110, 115, 是, 是], [[a1, a3, a4, s], 110, 0, 110, 115, 是, 是], [[a0, a1, a3, a4, s], 110, 0, 110, 115, 是, 是], [[a2, a3, a4, s], 110, 0, 110, 115, 是, 是], [[a0, a2, a3, a4, s], 110, 0, 110, 115, 是, 是], [[a1, a2, a3, a4, s], 110, 0, 110, 115, 是, 是], [[a0, a1, a2, a3, a4, s], 110, 0, 110, 115, 是, 是]], 是, 否, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum)；依完整明示資料獨立重算；feasible flow對每個cut皆有value≤capacity；同值flow/cut給雙optimality certificate；單一flow/cut不自動optimal；不計反向edges且不宣稱integrality、augmenting algorithm或一般定理已由instances證明。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r05-max-flow-min-cut-theorem-0045/)

最大流最小割定理 · qf-mod46-mfmc-flow-cut-upper-bound · candidate





## 46. 最大流最小割定理

完整有限network、全部s–t cuts與forward-edge cut capacity。操作=mfmc-flow-cut-upper-bound；資料={"vertices":["s","a0","a1","t"],"edges":[["s","a0"],["a0","t"],["s","a1"],["a1","t"]],"source":"s","sink":"t","capacities":[22,22,23,23],"flows":[21,21,22,22]}。請依 [source_value, sink_value, flow_cut_rows, all_cut_bounds_hold, automatic_maximum_claim, certificate] 欄序回答。

cut capacity只算向外邊；flow net是向外flow減向內flow。automatic_maximum_claim的否不是宣稱這個flow一定非最大。

查看答案與解釋(43, 43, [[[s], 43, 0, 43, 45, 是, 是], [[a0, s], 43, 0, 43, 45, 是, 是], [[a1, s], 43, 0, 43, 45, 是, 是], [[a0, a1, s], 43, 0, 43, 45, 是, 是]], 是, 否, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum)

Wave6第九層核對=(43, 43, [[[s], 43, 0, 43, 45, 是, 是], [[a0, s], 43, 0, 43, 45, 是, 是], [[a1, s], 43, 0, 43, 45, 是, 是], [[a0, a1, s], 43, 0, 43, 45, 是, 是]], 是, 否, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum)；依完整明示資料獨立重算；feasible flow對每個cut皆有value≤capacity；同值flow/cut給雙optimality certificate；單一flow/cut不自動optimal；不計反向edges且不宣稱integrality、augmenting algorithm或一般定理已由instances證明。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r05-max-flow-min-cut-theorem-0046/)

最大流最小割定理 · qf-mod46-mfmc-flow-cut-upper-bound · candidate





## 47. 最大流最小割定理

完整有限network、全部s–t cuts與forward-edge cut capacity。操作=mfmc-flow-cut-upper-bound；資料={"vertices":["s","a0","a1","a2","t"],"edges":[["s","a0"],["a0","t"],["s","a1"],["a1","t"],["s","a2"],["a2","t"],["a0","s"]],"source":"s","sink":"t","capacities":[23,23,24,24,25,25,28],"flows":[22,22,23,23,24,24,0]}。請依 [source_value, sink_value, flow_cut_rows, all_cut_bounds_hold, automatic_maximum_claim, certificate] 欄序回答。

cut capacity只算向外邊；flow net是向外flow減向內flow。automatic_maximum_claim的否不是宣稱這個flow一定非最大。

查看答案與解釋(69, 69, [[[s], 69, 0, 69, 72, 是, 是], [[a0, s], 69, 0, 69, 72, 是, 是], [[a1, s], 69, 0, 69, 72, 是, 是], [[a0, a1, s], 69, 0, 69, 72, 是, 是], [[a2, s], 69, 0, 69, 72, 是, 是], [[a0, a2, s], 69, 0, 69, 72, 是, 是], [[a1, a2, s], 69, 0, 69, 72, 是, 是], [[a0, a1, a2, s], 69, 0, 69, 72, 是, 是]], 是, 否, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum)

Wave6第九層核對=(69, 69, [[[s], 69, 0, 69, 72, 是, 是], [[a0, s], 69, 0, 69, 72, 是, 是], [[a1, s], 69, 0, 69, 72, 是, 是], [[a0, a1, s], 69, 0, 69, 72, 是, 是], [[a2, s], 69, 0, 69, 72, 是, 是], [[a0, a2, s], 69, 0, 69, 72, 是, 是], [[a1, a2, s], 69, 0, 69, 72, 是, 是], [[a0, a1, a2, s], 69, 0, 69, 72, 是, 是]], 是, 否, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum)；依完整明示資料獨立重算；feasible flow對每個cut皆有value≤capacity；同值flow/cut給雙optimality certificate；單一flow/cut不自動optimal；不計反向edges且不宣稱integrality、augmenting algorithm或一般定理已由instances證明。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r05-max-flow-min-cut-theorem-0047/)

最大流最小割定理 · qf-mod46-mfmc-flow-cut-upper-bound · candidate





## 48. 最大流最小割定理

完整有限network、全部s–t cuts與forward-edge cut capacity。操作=mfmc-flow-cut-upper-bound；資料={"vertices":["s","a0","a1","a2","a3","t"],"edges":[["s","a0"],["a0","t"],["s","a1"],["a1","t"],["s","a2"],["a2","t"],["s","a3"],["a3","t"]],"source":"s","sink":"t","capacities":[24,24,25,25,26,26,27,27],"flows":[23,23,24,24,25,25,26,26]}。請依 [source_value, sink_value, flow_cut_rows, all_cut_bounds_hold, automatic_maximum_claim, certificate] 欄序回答。

cut capacity只算向外邊；flow net是向外flow減向內flow。automatic_maximum_claim的否不是宣稱這個flow一定非最大。

查看答案與解釋(98, 98, [[[s], 98, 0, 98, 102, 是, 是], [[a0, s], 98, 0, 98, 102, 是, 是], [[a1, s], 98, 0, 98, 102, 是, 是], [[a0, a1, s], 98, 0, 98, 102, 是, 是], [[a2, s], 98, 0, 98, 102, 是, 是], [[a0, a2, s], 98, 0, 98, 102, 是, 是], [[a1, a2, s], 98, 0, 98, 102, 是, 是], [[a0, a1, a2, s], 98, 0, 98, 102, 是, 是], [[a3, s], 98, 0, 98, 102, 是, 是], [[a0, a3, s], 98, 0, 98, 102, 是, 是], [[a1, a3, s], 98, 0, 98, 102, 是, 是], [[a0, a1, a3, s], 98, 0, 98, 102, 是, 是], [[a2, a3, s], 98, 0, 98, 102, 是, 是], [[a0, a2, a3, s], 98, 0, 98, 102, 是, 是], [[a1, a2, a3, s], 98, 0, 98, 102, 是, 是], [[a0, a1, a2, a3, s], 98, 0, 98, 102, 是, 是]], 是, 否, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum)

Wave6第九層核對=(98, 98, [[[s], 98, 0, 98, 102, 是, 是], [[a0, s], 98, 0, 98, 102, 是, 是], [[a1, s], 98, 0, 98, 102, 是, 是], [[a0, a1, s], 98, 0, 98, 102, 是, 是], [[a2, s], 98, 0, 98, 102, 是, 是], [[a0, a2, s], 98, 0, 98, 102, 是, 是], [[a1, a2, s], 98, 0, 98, 102, 是, 是], [[a0, a1, a2, s], 98, 0, 98, 102, 是, 是], [[a3, s], 98, 0, 98, 102, 是, 是], [[a0, a3, s], 98, 0, 98, 102, 是, 是], [[a1, a3, s], 98, 0, 98, 102, 是, 是], [[a0, a1, a3, s], 98, 0, 98, 102, 是, 是], [[a2, a3, s], 98, 0, 98, 102, 是, 是], [[a0, a2, a3, s], 98, 0, 98, 102, 是, 是], [[a1, a2, a3, s], 98, 0, 98, 102, 是, 是], [[a0, a1, a2, a3, s], 98, 0, 98, 102, 是, 是]], 是, 否, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum)；依完整明示資料獨立重算；feasible flow對每個cut皆有value≤capacity；同值flow/cut給雙optimality certificate；單一flow/cut不自動optimal；不計反向edges且不宣稱integrality、augmenting algorithm或一般定理已由instances證明。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r05-max-flow-min-cut-theorem-0048/)

最大流最小割定理 · qf-mod46-mfmc-flow-cut-upper-bound · candidate





## 49. 最大流最小割定理

完整有限network、全部s–t cuts與forward-edge cut capacity。操作=mfmc-flow-cut-upper-bound；資料={"vertices":["s","a0","a1","a2","a3","a4","t"],"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","capacities":[25,25,26,26,27,27,28,28,29,29],"flows":[24,24,25,25,26,26,27,27,28,28]}。請依 [source_value, sink_value, flow_cut_rows, all_cut_bounds_hold, automatic_maximum_claim, certificate] 欄序回答。

cut capacity只算向外邊；flow net是向外flow減向內flow。automatic_maximum_claim的否不是宣稱這個flow一定非最大。

查看答案與解釋(130, 130, [[[s], 130, 0, 130, 135, 是, 是], [[a0, s], 130, 0, 130, 135, 是, 是], [[a1, s], 130, 0, 130, 135, 是, 是], [[a0, a1, s], 130, 0, 130, 135, 是, 是], [[a2, s], 130, 0, 130, 135, 是, 是], [[a0, a2, s], 130, 0, 130, 135, 是, 是], [[a1, a2, s], 130, 0, 130, 135, 是, 是], [[a0, a1, a2, s], 130, 0, 130, 135, 是, 是], [[a3, s], 130, 0, 130, 135, 是, 是], [[a0, a3, s], 130, 0, 130, 135, 是, 是], [[a1, a3, s], 130, 0, 130, 135, 是, 是], [[a0, a1, a3, s], 130, 0, 130, 135, 是, 是], [[a2, a3, s], 130, 0, 130, 135, 是, 是], [[a0, a2, a3, s], 130, 0, 130, 135, 是, 是], [[a1, a2, a3, s], 130, 0, 130, 135, 是, 是], [[a0, a1, a2, a3, s], 130, 0, 130, 135, 是, 是], [[a4, s], 130, 0, 130, 135, 是, 是], [[a0, a4, s], 130, 0, 130, 135, 是, 是], [[a1, a4, s], 130, 0, 130, 135, 是, 是], [[a0, a1, a4, s], 130, 0, 130, 135, 是, 是], [[a2, a4, s], 130, 0, 130, 135, 是, 是], [[a0, a2, a4, s], 130, 0, 130, 135, 是, 是], [[a1, a2, a4, s], 130, 0, 130, 135, 是, 是], [[a0, a1, a2, a4, s], 130, 0, 130, 135, 是, 是], [[a3, a4, s], 130, 0, 130, 135, 是, 是], [[a0, a3, a4, s], 130, 0, 130, 135, 是, 是], [[a1, a3, a4, s], 130, 0, 130, 135, 是, 是], [[a0, a1, a3, a4, s], 130, 0, 130, 135, 是, 是], [[a2, a3, a4, s], 130, 0, 130, 135, 是, 是], [[a0, a2, a3, a4, s], 130, 0, 130, 135, 是, 是], [[a1, a2, a3, a4, s], 130, 0, 130, 135, 是, 是], [[a0, a1, a2, a3, a4, s], 130, 0, 130, 135, 是, 是]], 是, 否, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum)

Wave6第九層核對=(130, 130, [[[s], 130, 0, 130, 135, 是, 是], [[a0, s], 130, 0, 130, 135, 是, 是], [[a1, s], 130, 0, 130, 135, 是, 是], [[a0, a1, s], 130, 0, 130, 135, 是, 是], [[a2, s], 130, 0, 130, 135, 是, 是], [[a0, a2, s], 130, 0, 130, 135, 是, 是], [[a1, a2, s], 130, 0, 130, 135, 是, 是], [[a0, a1, a2, s], 130, 0, 130, 135, 是, 是], [[a3, s], 130, 0, 130, 135, 是, 是], [[a0, a3, s], 130, 0, 130, 135, 是, 是], [[a1, a3, s], 130, 0, 130, 135, 是, 是], [[a0, a1, a3, s], 130, 0, 130, 135, 是, 是], [[a2, a3, s], 130, 0, 130, 135, 是, 是], [[a0, a2, a3, s], 130, 0, 130, 135, 是, 是], [[a1, a2, a3, s], 130, 0, 130, 135, 是, 是], [[a0, a1, a2, a3, s], 130, 0, 130, 135, 是, 是], [[a4, s], 130, 0, 130, 135, 是, 是], [[a0, a4, s], 130, 0, 130, 135, 是, 是], [[a1, a4, s], 130, 0, 130, 135, 是, 是], [[a0, a1, a4, s], 130, 0, 130, 135, 是, 是], [[a2, a4, s], 130, 0, 130, 135, 是, 是], [[a0, a2, a4, s], 130, 0, 130, 135, 是, 是], [[a1, a2, a4, s], 130, 0, 130, 135, 是, 是], [[a0, a1, a2, a4, s], 130, 0, 130, 135, 是, 是], [[a3, a4, s], 130, 0, 130, 135, 是, 是], [[a0, a3, a4, s], 130, 0, 130, 135, 是, 是], [[a1, a3, a4, s], 130, 0, 130, 135, 是, 是], [[a0, a1, a3, a4, s], 130, 0, 130, 135, 是, 是], [[a2, a3, a4, s], 130, 0, 130, 135, 是, 是], [[a0, a2, a3, a4, s], 130, 0, 130, 135, 是, 是], [[a1, a2, a3, a4, s], 130, 0, 130, 135, 是, 是], [[a0, a1, a2, a3, a4, s], 130, 0, 130, 135, 是, 是]], 是, 否, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum)；依完整明示資料獨立重算；feasible flow對每個cut皆有value≤capacity；同值flow/cut給雙optimality certificate；單一flow/cut不自動optimal；不計反向edges且不宣稱integrality、augmenting algorithm或一般定理已由instances證明。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r05-max-flow-min-cut-theorem-0049/)

最大流最小割定理 · qf-mod46-mfmc-flow-cut-upper-bound · candidate





## 50. 最大流最小割定理

完整有限network、全部s–t cuts與forward-edge cut capacity。操作=mfmc-flow-cut-upper-bound；資料={"vertices":["s","a0","a1","t"],"edges":[["s","a0"],["a0","t"],["s","a1"],["a1","t"],["a0","s"]],"source":"s","sink":"t","capacities":[26,26,27,27,31],"flows":[25,25,26,26,0]}。請依 [source_value, sink_value, flow_cut_rows, all_cut_bounds_hold, automatic_maximum_claim, certificate] 欄序回答。

cut capacity只算向外邊；flow net是向外flow減向內flow。automatic_maximum_claim的否不是宣稱這個flow一定非最大。

查看答案與解釋(51, 51, [[[s], 51, 0, 51, 53, 是, 是], [[a0, s], 51, 0, 51, 53, 是, 是], [[a1, s], 51, 0, 51, 53, 是, 是], [[a0, a1, s], 51, 0, 51, 53, 是, 是]], 是, 否, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum)

Wave6第九層核對=(51, 51, [[[s], 51, 0, 51, 53, 是, 是], [[a0, s], 51, 0, 51, 53, 是, 是], [[a1, s], 51, 0, 51, 53, 是, 是], [[a0, a1, s], 51, 0, 51, 53, 是, 是]], 是, 否, every_feasible_flow_value_is_at_most_every_cut_capacity_but_need_not_be_maximum)；依完整明示資料獨立重算；feasible flow對每個cut皆有value≤capacity；同值flow/cut給雙optimality certificate；單一flow/cut不自動optimal；不計反向edges且不宣稱integrality、augmenting algorithm或一般定理已由instances證明。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r05-max-flow-min-cut-theorem-0050/)

最大流最小割定理 · qf-mod46-mfmc-flow-cut-upper-bound · candidate





## 51. 最大流最小割定理

完整有限network、全部s–t cuts與forward-edge cut capacity。操作=mfmc-minimum-cut-audit；資料={"vertices":["s","a0","a1","t"],"edges":[["s","a0"],["a0","t"],["s","a1"],["a1","t"],["a0","s"]],"source":"s","sink":"t","capacities":[2,2,3,3,7]}。請依 [cut_rows, minimum_capacity, minimum_cuts, certificate] 欄序回答。

查看答案與解釋([[[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)

Wave6第九層核對=([[[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)；依完整明示資料獨立重算；feasible flow對每個cut皆有value≤capacity；同值flow/cut給雙optimality certificate；單一flow/cut不自動optimal；不計反向edges且不宣稱integrality、augmenting algorithm或一般定理已由instances證明。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r05-max-flow-min-cut-theorem-0051/)

最大流最小割定理 · qf-mod46-mfmc-minimum-cut-audit · candidate





## 52. 最大流最小割定理

完整有限network、全部s–t cuts與forward-edge cut capacity。操作=mfmc-minimum-cut-audit；資料={"vertices":["s","a0","a1","a2","t"],"edges":[["s","a0"],["a0","t"],["s","a1"],["a1","t"],["s","a2"],["a2","t"]],"source":"s","sink":"t","capacities":[3,3,4,4,5,5]}。請依 [cut_rows, minimum_capacity, minimum_cuts, certificate] 欄序回答。

查看答案與解釋([[[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)

Wave6第九層核對=([[[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)；依完整明示資料獨立重算；feasible flow對每個cut皆有value≤capacity；同值flow/cut給雙optimality certificate；單一flow/cut不自動optimal；不計反向edges且不宣稱integrality、augmenting algorithm或一般定理已由instances證明。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r05-max-flow-min-cut-theorem-0052/)

最大流最小割定理 · qf-mod46-mfmc-minimum-cut-audit · candidate





## 53. 最大流最小割定理

完整有限network、全部s–t cuts與forward-edge cut capacity。操作=mfmc-minimum-cut-audit；資料={"vertices":["s","a0","a1","a2","a3","t"],"edges":[["s","a0"],["a0","t"],["s","a1"],["a1","t"],["s","a2"],["a2","t"],["s","a3"],["a3","t"]],"source":"s","sink":"t","capacities":[4,4,5,5,6,6,7,7]}。請依 [cut_rows, minimum_capacity, minimum_cuts, certificate] 欄序回答。

查看答案與解釋([[[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)

Wave6第九層核對=([[[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)；依完整明示資料獨立重算；feasible flow對每個cut皆有value≤capacity；同值flow/cut給雙optimality certificate；單一flow/cut不自動optimal；不計反向edges且不宣稱integrality、augmenting algorithm或一般定理已由instances證明。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r05-max-flow-min-cut-theorem-0053/)

最大流最小割定理 · qf-mod46-mfmc-minimum-cut-audit · candidate





## 54. 最大流最小割定理

完整有限network、全部s–t cuts與forward-edge cut capacity。操作=mfmc-minimum-cut-audit；資料={"vertices":["s","a0","a1","a2","a3","a4","t"],"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","capacities":[5,5,6,6,7,7,8,8,9,9,10]}。請依 [cut_rows, minimum_capacity, minimum_cuts, certificate] 欄序回答。

查看答案與解釋([[[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)

Wave6第九層核對=([[[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)；依完整明示資料獨立重算；feasible flow對每個cut皆有value≤capacity；同值flow/cut給雙optimality certificate；單一flow/cut不自動optimal；不計反向edges且不宣稱integrality、augmenting algorithm或一般定理已由instances證明。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r05-max-flow-min-cut-theorem-0054/)

最大流最小割定理 · qf-mod46-mfmc-minimum-cut-audit · candidate





## 55. 最大流最小割定理

完整有限network、全部s–t cuts與forward-edge cut capacity。操作=mfmc-minimum-cut-audit；資料={"vertices":["s","a0","a1","t"],"edges":[["s","a0"],["a0","t"],["s","a1"],["a1","t"]],"source":"s","sink":"t","capacities":[6,6,7,7]}。請依 [cut_rows, minimum_capacity, minimum_cuts, certificate] 欄序回答。

查看答案與解釋([[[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)

Wave6第九層核對=([[[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)；依完整明示資料獨立重算；feasible flow對每個cut皆有value≤capacity；同值flow/cut給雙optimality certificate；單一flow/cut不自動optimal；不計反向edges且不宣稱integrality、augmenting algorithm或一般定理已由instances證明。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r05-max-flow-min-cut-theorem-0055/)

最大流最小割定理 · qf-mod46-mfmc-minimum-cut-audit · candidate





## 56. 最大流最小割定理

完整有限network、全部s–t cuts與forward-edge cut capacity。操作=mfmc-minimum-cut-audit；資料={"vertices":["s","a0","a1","a2","t"],"edges":[["s","a0"],["a0","t"],["s","a1"],["a1","t"],["s","a2"],["a2","t"]],"source":"s","sink":"t","capacities":[7,7,8,8,9,9]}。請依 [cut_rows, minimum_capacity, minimum_cuts, certificate] 欄序回答。

查看答案與解釋([[[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)

Wave6第九層核對=([[[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)；依完整明示資料獨立重算；feasible flow對每個cut皆有value≤capacity；同值flow/cut給雙optimality certificate；單一flow/cut不自動optimal；不計反向edges且不宣稱integrality、augmenting algorithm或一般定理已由instances證明。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r05-max-flow-min-cut-theorem-0056/)

最大流最小割定理 · qf-mod46-mfmc-minimum-cut-audit · candidate





## 57. 最大流最小割定理

完整有限network、全部s–t cuts與forward-edge cut capacity。操作=mfmc-minimum-cut-audit；資料={"vertices":["s","a0","a1","a2","a3","t"],"edges":[["s","a0"],["a0","t"],["s","a1"],["a1","t"],["s","a2"],["a2","t"],["s","a3"],["a3","t"],["a0","s"]],"source":"s","sink":"t","capacities":[8,8,9,9,10,10,11,11,13]}。請依 [cut_rows, minimum_capacity, minimum_cuts, certificate] 欄序回答。

查看答案與解釋([[[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)

Wave6第九層核對=([[[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)；依完整明示資料獨立重算；feasible flow對每個cut皆有value≤capacity；同值flow/cut給雙optimality certificate；單一flow/cut不自動optimal；不計反向edges且不宣稱integrality、augmenting algorithm或一般定理已由instances證明。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r05-max-flow-min-cut-theorem-0057/)

最大流最小割定理 · qf-mod46-mfmc-minimum-cut-audit · candidate





## 58. 最大流最小割定理

完整有限network、全部s–t cuts與forward-edge cut capacity。操作=mfmc-minimum-cut-audit；資料={"vertices":["s","a0","a1","a2","a3","a4","t"],"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","capacities":[9,9,10,10,11,11,12,12,13,13]}。請依 [cut_rows, minimum_capacity, minimum_cuts, certificate] 欄序回答。

查看答案與解釋([[[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)

Wave6第九層核對=([[[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)；依完整明示資料獨立重算；feasible flow對每個cut皆有value≤capacity；同值flow/cut給雙optimality certificate；單一flow/cut不自動optimal；不計反向edges且不宣稱integrality、augmenting algorithm或一般定理已由instances證明。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r05-max-flow-min-cut-theorem-0058/)

最大流最小割定理 · qf-mod46-mfmc-minimum-cut-audit · candidate





## 59. 最大流最小割定理

完整有限network、全部s–t cuts與forward-edge cut capacity。操作=mfmc-minimum-cut-audit；資料={"vertices":["s","a0","a1","t"],"edges":[["s","a0"],["a0","t"],["s","a1"],["a1","t"]],"source":"s","sink":"t","capacities":[10,10,11,11]}。請依 [cut_rows, minimum_capacity, minimum_cuts, certificate] 欄序回答。

查看答案與解釋([[[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)

Wave6第九層核對=([[[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)；依完整明示資料獨立重算；feasible flow對每個cut皆有value≤capacity；同值flow/cut給雙optimality certificate；單一flow/cut不自動optimal；不計反向edges且不宣稱integrality、augmenting algorithm或一般定理已由instances證明。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r05-max-flow-min-cut-theorem-0059/)

最大流最小割定理 · qf-mod46-mfmc-minimum-cut-audit · candidate





## 60. 最大流最小割定理

完整有限network、全部s–t cuts與forward-edge cut capacity。操作=mfmc-minimum-cut-audit；資料={"vertices":["s","a0","a1","a2","t"],"edges":[["s","a0"],["a0","t"],["s","a1"],["a1","t"],["s","a2"],["a2","t"],["a0","s"]],"source":"s","sink":"t","capacities":[11,11,12,12,13,13,16]}。請依 [cut_rows, minimum_capacity, minimum_cuts, certificate] 欄序回答。

查看答案與解釋([[[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)

Wave6第九層核對=([[[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)；依完整明示資料獨立重算；feasible flow對每個cut皆有value≤capacity；同值flow/cut給雙optimality certificate；單一flow/cut不自動optimal；不計反向edges且不宣稱integrality、augmenting algorithm或一般定理已由instances證明。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r05-max-flow-min-cut-theorem-0060/)

最大流最小割定理 · qf-mod46-mfmc-minimum-cut-audit · candidate





## 61. 最大流最小割定理

完整有限network、全部s–t cuts與forward-edge cut capacity。操作=mfmc-minimum-cut-audit；資料={"vertices":["s","a0","a1","a2","a3","t"],"edges":[["s","a0"],["a0","t"],["s","a1"],["a1","t"],["s","a2"],["a2","t"],["s","a3"],["a3","t"]],"source":"s","sink":"t","capacities":[12,12,13,13,14,14,15,15]}。請依 [cut_rows, minimum_capacity, minimum_cuts, certificate] 欄序回答。

查看答案與解釋([[[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)

Wave6第九層核對=([[[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)；依完整明示資料獨立重算；feasible flow對每個cut皆有value≤capacity；同值flow/cut給雙optimality certificate；單一flow/cut不自動optimal；不計反向edges且不宣稱integrality、augmenting algorithm或一般定理已由instances證明。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r05-max-flow-min-cut-theorem-0061/)

最大流最小割定理 · qf-mod46-mfmc-minimum-cut-audit · candidate





## 62. 最大流最小割定理

完整有限network、全部s–t cuts與forward-edge cut capacity。操作=mfmc-minimum-cut-audit；資料={"vertices":["s","a0","a1","a2","a3","a4","t"],"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","capacities":[13,13,14,14,15,15,16,16,17,17]}。請依 [cut_rows, minimum_capacity, minimum_cuts, certificate] 欄序回答。

查看答案與解釋([[[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)

Wave6第九層核對=([[[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)；依完整明示資料獨立重算；feasible flow對每個cut皆有value≤capacity；同值flow/cut給雙optimality certificate；單一flow/cut不自動optimal；不計反向edges且不宣稱integrality、augmenting algorithm或一般定理已由instances證明。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r05-max-flow-min-cut-theorem-0062/)

最大流最小割定理 · qf-mod46-mfmc-minimum-cut-audit · candidate





## 63. 最大流最小割定理

完整有限network、全部s–t cuts與forward-edge cut capacity。操作=mfmc-minimum-cut-audit；資料={"vertices":["s","a0","a1","t"],"edges":[["s","a0"],["a0","t"],["s","a1"],["a1","t"],["a0","s"]],"source":"s","sink":"t","capacities":[14,14,15,15,19]}。請依 [cut_rows, minimum_capacity, minimum_cuts, certificate] 欄序回答。

查看答案與解釋([[[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)

Wave6第九層核對=([[[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)；依完整明示資料獨立重算；feasible flow對每個cut皆有value≤capacity；同值flow/cut給雙optimality certificate；單一flow/cut不自動optimal；不計反向edges且不宣稱integrality、augmenting algorithm或一般定理已由instances證明。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r05-max-flow-min-cut-theorem-0063/)

最大流最小割定理 · qf-mod46-mfmc-minimum-cut-audit · candidate





## 64. 最大流最小割定理

完整有限network、全部s–t cuts與forward-edge cut capacity。操作=mfmc-minimum-cut-audit；資料={"vertices":["s","a0","a1","a2","t"],"edges":[["s","a0"],["a0","t"],["s","a1"],["a1","t"],["s","a2"],["a2","t"]],"source":"s","sink":"t","capacities":[15,15,16,16,17,17]}。請依 [cut_rows, minimum_capacity, minimum_cuts, certificate] 欄序回答。

查看答案與解釋([[[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)

Wave6第九層核對=([[[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)；依完整明示資料獨立重算；feasible flow對每個cut皆有value≤capacity；同值flow/cut給雙optimality certificate；單一flow/cut不自動optimal；不計反向edges且不宣稱integrality、augmenting algorithm或一般定理已由instances證明。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r05-max-flow-min-cut-theorem-0064/)

最大流最小割定理 · qf-mod46-mfmc-minimum-cut-audit · candidate





## 65. 最大流最小割定理

完整有限network、全部s–t cuts與forward-edge cut capacity。操作=mfmc-minimum-cut-audit；資料={"vertices":["s","a0","a1","a2","a3","t"],"edges":[["s","a0"],["a0","t"],["s","a1"],["a1","t"],["s","a2"],["a2","t"],["s","a3"],["a3","t"]],"source":"s","sink":"t","capacities":[16,16,17,17,18,18,19,19]}。請依 [cut_rows, minimum_capacity, minimum_cuts, certificate] 欄序回答。

查看答案與解釋([[[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)

Wave6第九層核對=([[[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)；依完整明示資料獨立重算；feasible flow對每個cut皆有value≤capacity；同值flow/cut給雙optimality certificate；單一flow/cut不自動optimal；不計反向edges且不宣稱integrality、augmenting algorithm或一般定理已由instances證明。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r05-max-flow-min-cut-theorem-0065/)

最大流最小割定理 · qf-mod46-mfmc-minimum-cut-audit · candidate





## 66. 最大流最小割定理

完整有限network、全部s–t cuts與forward-edge cut capacity。操作=mfmc-minimum-cut-audit；資料={"vertices":["s","a0","a1","a2","a3","a4","t"],"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","capacities":[17,17,18,18,19,19,20,20,21,21,22]}。請依 [cut_rows, minimum_capacity, minimum_cuts, certificate] 欄序回答。

查看答案與解釋([[[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)

Wave6第九層核對=([[[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)；依完整明示資料獨立重算；feasible flow對每個cut皆有value≤capacity；同值flow/cut給雙optimality certificate；單一flow/cut不自動optimal；不計反向edges且不宣稱integrality、augmenting algorithm或一般定理已由instances證明。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r05-max-flow-min-cut-theorem-0066/)

最大流最小割定理 · qf-mod46-mfmc-minimum-cut-audit · candidate





## 67. 最大流最小割定理

完整有限network、全部s–t cuts與forward-edge cut capacity。操作=mfmc-minimum-cut-audit；資料={"vertices":["s","a0","a1","t"],"edges":[["s","a0"],["a0","t"],["s","a1"],["a1","t"]],"source":"s","sink":"t","capacities":[18,18,19,19]}。請依 [cut_rows, minimum_capacity, minimum_cuts, certificate] 欄序回答。

查看答案與解釋([[[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)

Wave6第九層核對=([[[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)；依完整明示資料獨立重算；feasible flow對每個cut皆有value≤capacity；同值flow/cut給雙optimality certificate；單一flow/cut不自動optimal；不計反向edges且不宣稱integrality、augmenting algorithm或一般定理已由instances證明。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r05-max-flow-min-cut-theorem-0067/)

最大流最小割定理 · qf-mod46-mfmc-minimum-cut-audit · candidate





## 68. 最大流最小割定理

完整有限network、全部s–t cuts與forward-edge cut capacity。操作=mfmc-minimum-cut-audit；資料={"vertices":["s","a0","a1","a2","t"],"edges":[["s","a0"],["a0","t"],["s","a1"],["a1","t"],["s","a2"],["a2","t"]],"source":"s","sink":"t","capacities":[19,19,20,20,21,21]}。請依 [cut_rows, minimum_capacity, minimum_cuts, certificate] 欄序回答。

查看答案與解釋([[[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)

Wave6第九層核對=([[[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)；依完整明示資料獨立重算；feasible flow對每個cut皆有value≤capacity；同值flow/cut給雙optimality certificate；單一flow/cut不自動optimal；不計反向edges且不宣稱integrality、augmenting algorithm或一般定理已由instances證明。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r05-max-flow-min-cut-theorem-0068/)

最大流最小割定理 · qf-mod46-mfmc-minimum-cut-audit · candidate





## 69. 最大流最小割定理

完整有限network、全部s–t cuts與forward-edge cut capacity。操作=mfmc-minimum-cut-audit；資料={"vertices":["s","a0","a1","a2","a3","t"],"edges":[["s","a0"],["a0","t"],["s","a1"],["a1","t"],["s","a2"],["a2","t"],["s","a3"],["a3","t"],["a0","s"]],"source":"s","sink":"t","capacities":[20,20,21,21,22,22,23,23,25]}。請依 [cut_rows, minimum_capacity, minimum_cuts, certificate] 欄序回答。

查看答案與解釋([[[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)

Wave6第九層核對=([[[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)；依完整明示資料獨立重算；feasible flow對每個cut皆有value≤capacity；同值flow/cut給雙optimality certificate；單一flow/cut不自動optimal；不計反向edges且不宣稱integrality、augmenting algorithm或一般定理已由instances證明。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r05-max-flow-min-cut-theorem-0069/)

最大流最小割定理 · qf-mod46-mfmc-minimum-cut-audit · candidate





## 70. 最大流最小割定理

完整有限network、全部s–t cuts與forward-edge cut capacity。操作=mfmc-minimum-cut-audit；資料={"vertices":["s","a0","a1","a2","a3","a4","t"],"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","capacities":[21,21,22,22,23,23,24,24,25,25]}。請依 [cut_rows, minimum_capacity, minimum_cuts, certificate] 欄序回答。

查看答案與解釋([[[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)

Wave6第九層核對=([[[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)；依完整明示資料獨立重算；feasible flow對每個cut皆有value≤capacity；同值flow/cut給雙optimality certificate；單一flow/cut不自動optimal；不計反向edges且不宣稱integrality、augmenting algorithm或一般定理已由instances證明。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r05-max-flow-min-cut-theorem-0070/)

最大流最小割定理 · qf-mod46-mfmc-minimum-cut-audit · candidate





## 71. 最大流最小割定理

完整有限network、全部s–t cuts與forward-edge cut capacity。操作=mfmc-minimum-cut-audit；資料={"vertices":["s","a0","a1","t"],"edges":[["s","a0"],["a0","t"],["s","a1"],["a1","t"]],"source":"s","sink":"t","capacities":[22,22,23,23]}。請依 [cut_rows, minimum_capacity, minimum_cuts, certificate] 欄序回答。

查看答案與解釋([[[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)

Wave6第九層核對=([[[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)；依完整明示資料獨立重算；feasible flow對每個cut皆有value≤capacity；同值flow/cut給雙optimality certificate；單一flow/cut不自動optimal；不計反向edges且不宣稱integrality、augmenting algorithm或一般定理已由instances證明。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r05-max-flow-min-cut-theorem-0071/)

最大流最小割定理 · qf-mod46-mfmc-minimum-cut-audit · candidate





## 72. 最大流最小割定理

完整有限network、全部s–t cuts與forward-edge cut capacity。操作=mfmc-minimum-cut-audit；資料={"vertices":["s","a0","a1","a2","t"],"edges":[["s","a0"],["a0","t"],["s","a1"],["a1","t"],["s","a2"],["a2","t"],["a0","s"]],"source":"s","sink":"t","capacities":[23,23,24,24,25,25,28]}。請依 [cut_rows, minimum_capacity, minimum_cuts, certificate] 欄序回答。

查看答案與解釋([[[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)

Wave6第九層核對=([[[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)；依完整明示資料獨立重算；feasible flow對每個cut皆有value≤capacity；同值flow/cut給雙optimality certificate；單一flow/cut不自動optimal；不計反向edges且不宣稱integrality、augmenting algorithm或一般定理已由instances證明。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r05-max-flow-min-cut-theorem-0072/)

最大流最小割定理 · qf-mod46-mfmc-minimum-cut-audit · candidate





## 73. 最大流最小割定理

完整有限network、全部s–t cuts與forward-edge cut capacity。操作=mfmc-minimum-cut-audit；資料={"vertices":["s","a0","a1","a2","a3","t"],"edges":[["s","a0"],["a0","t"],["s","a1"],["a1","t"],["s","a2"],["a2","t"],["s","a3"],["a3","t"]],"source":"s","sink":"t","capacities":[24,24,25,25,26,26,27,27]}。請依 [cut_rows, minimum_capacity, minimum_cuts, certificate] 欄序回答。

查看答案與解釋([[[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)

Wave6第九層核對=([[[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)；依完整明示資料獨立重算；feasible flow對每個cut皆有value≤capacity；同值flow/cut給雙optimality certificate；單一flow/cut不自動optimal；不計反向edges且不宣稱integrality、augmenting algorithm或一般定理已由instances證明。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r05-max-flow-min-cut-theorem-0073/)

最大流最小割定理 · qf-mod46-mfmc-minimum-cut-audit · candidate





## 74. 最大流最小割定理

完整有限network、全部s–t cuts與forward-edge cut capacity。操作=mfmc-minimum-cut-audit；資料={"vertices":["s","a0","a1","a2","a3","a4","t"],"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","capacities":[25,25,26,26,27,27,28,28,29,29]}。請依 [cut_rows, minimum_capacity, minimum_cuts, certificate] 欄序回答。

查看答案與解釋([[[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)

Wave6第九層核對=([[[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)；依完整明示資料獨立重算；feasible flow對每個cut皆有value≤capacity；同值flow/cut給雙optimality certificate；單一flow/cut不自動optimal；不計反向edges且不宣稱integrality、augmenting algorithm或一般定理已由instances證明。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r05-max-flow-min-cut-theorem-0074/)

最大流最小割定理 · qf-mod46-mfmc-minimum-cut-audit · candidate





## 75. 最大流最小割定理

完整有限network、全部s–t cuts與forward-edge cut capacity。操作=mfmc-minimum-cut-audit；資料={"vertices":["s","a0","a1","t"],"edges":[["s","a0"],["a0","t"],["s","a1"],["a1","t"],["a0","s"]],"source":"s","sink":"t","capacities":[26,26,27,27,31]}。請依 [cut_rows, minimum_capacity, minimum_cuts, certificate] 欄序回答。

查看答案與解釋([[[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)

Wave6第九層核對=([[[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)；依完整明示資料獨立重算；feasible flow對每個cut皆有value≤capacity；同值flow/cut給雙optimality certificate；單一flow/cut不自動optimal；不計反向edges且不宣稱integrality、augmenting algorithm或一般定理已由instances證明。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r05-max-flow-min-cut-theorem-0075/)

最大流最小割定理 · qf-mod46-mfmc-minimum-cut-audit · candidate





## 76. 最大流最小割定理

完整有限network、全部s–t cuts與forward-edge cut capacity。操作=mfmc-equal-value-optimality；資料={"vertices":["s","a0","a1","t"],"edges":[["s","a0"],["a0","t"],["s","a1"],["a1","t"],["a0","s"]],"source":"s","sink":"t","capacities":[2,2,3,3,7],"flows":[2,2,3,3,0]}。請依 [flow_value, minimum_capacity, equal_value_cuts, minimum_cuts, equal_values, maximum_flow_certificate, minimum_cut_certificate, integrality_or_algorithm_claim, certificate] 欄序回答。

同值證書先要求feasible flow；最後否只不提供integrality或特定algorithm證明，不否定maximum/minimum。

查看答案與解釋(5, 5, [[s], [a0, s], [a1, s], [a0, a1, s]], [[s], [a0, s], [a1, s], [a0, a1, s]], 是, 是, 是, 否, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims)

Wave6第九層核對=(5, 5, [[s], [a0, s], [a1, s], [a0, a1, s]], [[s], [a0, s], [a1, s], [a0, a1, s]], 是, 是, 是, 否, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims)；依完整明示資料獨立重算；feasible flow對每個cut皆有value≤capacity；同值flow/cut給雙optimality certificate；單一flow/cut不自動optimal；不計反向edges且不宣稱integrality、augmenting algorithm或一般定理已由instances證明。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r05-max-flow-min-cut-theorem-0076/)

最大流最小割定理 · qf-mod46-mfmc-equal-value-optimality · candidate





## 77. 最大流最小割定理

完整有限network、全部s–t cuts與forward-edge cut capacity。操作=mfmc-equal-value-optimality；資料={"vertices":["s","a0","a1","a2","t"],"edges":[["s","a0"],["a0","t"],["s","a1"],["a1","t"],["s","a2"],["a2","t"]],"source":"s","sink":"t","capacities":[3,3,4,4,5,5],"flows":[3,3,4,4,5,5]}。請依 [flow_value, minimum_capacity, equal_value_cuts, minimum_cuts, equal_values, maximum_flow_certificate, minimum_cut_certificate, integrality_or_algorithm_claim, certificate] 欄序回答。

同值證書先要求feasible flow；最後否只不提供integrality或特定algorithm證明，不否定maximum/minimum。

查看答案與解釋(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]], 是, 是, 是, 否, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims)

Wave6第九層核對=(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]], 是, 是, 是, 否, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims)；依完整明示資料獨立重算；feasible flow對每個cut皆有value≤capacity；同值flow/cut給雙optimality certificate；單一flow/cut不自動optimal；不計反向edges且不宣稱integrality、augmenting algorithm或一般定理已由instances證明。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r05-max-flow-min-cut-theorem-0077/)

最大流最小割定理 · qf-mod46-mfmc-equal-value-optimality · candidate





## 78. 最大流最小割定理

完整有限network、全部s–t cuts與forward-edge cut capacity。操作=mfmc-equal-value-optimality；資料={"vertices":["s","a0","a1","a2","a3","t"],"edges":[["s","a0"],["a0","t"],["s","a1"],["a1","t"],["s","a2"],["a2","t"],["s","a3"],["a3","t"]],"source":"s","sink":"t","capacities":[4,4,5,5,6,6,7,7],"flows":[4,4,5,5,6,6,7,7]}。請依 [flow_value, minimum_capacity, equal_value_cuts, minimum_cuts, equal_values, maximum_flow_certificate, minimum_cut_certificate, integrality_or_algorithm_claim, certificate] 欄序回答。

同值證書先要求feasible flow；最後否只不提供integrality或特定algorithm證明，不否定maximum/minimum。

查看答案與解釋(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]], 是, 是, 是, 否, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims)

Wave6第九層核對=(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]], 是, 是, 是, 否, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims)；依完整明示資料獨立重算；feasible flow對每個cut皆有value≤capacity；同值flow/cut給雙optimality certificate；單一flow/cut不自動optimal；不計反向edges且不宣稱integrality、augmenting algorithm或一般定理已由instances證明。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r05-max-flow-min-cut-theorem-0078/)

最大流最小割定理 · qf-mod46-mfmc-equal-value-optimality · candidate





## 79. 最大流最小割定理

完整有限network、全部s–t cuts與forward-edge cut capacity。操作=mfmc-equal-value-optimality；資料={"vertices":["s","a0","a1","a2","a3","a4","t"],"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","capacities":[5,5,6,6,7,7,8,8,9,9,10],"flows":[5,5,6,6,7,7,8,8,9,9,0]}。請依 [flow_value, minimum_capacity, equal_value_cuts, minimum_cuts, equal_values, maximum_flow_certificate, minimum_cut_certificate, integrality_or_algorithm_claim, certificate] 欄序回答。

同值證書先要求feasible flow；最後否只不提供integrality或特定algorithm證明，不否定maximum/minimum。

查看答案與解釋(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]], 是, 是, 是, 否, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims)

Wave6第九層核對=(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]], 是, 是, 是, 否, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims)；依完整明示資料獨立重算；feasible flow對每個cut皆有value≤capacity；同值flow/cut給雙optimality certificate；單一flow/cut不自動optimal；不計反向edges且不宣稱integrality、augmenting algorithm或一般定理已由instances證明。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r05-max-flow-min-cut-theorem-0079/)

最大流最小割定理 · qf-mod46-mfmc-equal-value-optimality · candidate





## 80. 最大流最小割定理

完整有限network、全部s–t cuts與forward-edge cut capacity。操作=mfmc-equal-value-optimality；資料={"vertices":["s","a0","a1","t"],"edges":[["s","a0"],["a0","t"],["s","a1"],["a1","t"]],"source":"s","sink":"t","capacities":[6,6,7,7],"flows":[6,6,7,7]}。請依 [flow_value, minimum_capacity, equal_value_cuts, minimum_cuts, equal_values, maximum_flow_certificate, minimum_cut_certificate, integrality_or_algorithm_claim, certificate] 欄序回答。

同值證書先要求feasible flow；最後否只不提供integrality或特定algorithm證明，不否定maximum/minimum。

查看答案與解釋(13, 13, [[s], [a0, s], [a1, s], [a0, a1, s]], [[s], [a0, s], [a1, s], [a0, a1, s]], 是, 是, 是, 否, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims)

Wave6第九層核對=(13, 13, [[s], [a0, s], [a1, s], [a0, a1, s]], [[s], [a0, s], [a1, s], [a0, a1, s]], 是, 是, 是, 否, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims)；依完整明示資料獨立重算；feasible flow對每個cut皆有value≤capacity；同值flow/cut給雙optimality certificate；單一flow/cut不自動optimal；不計反向edges且不宣稱integrality、augmenting algorithm或一般定理已由instances證明。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r05-max-flow-min-cut-theorem-0080/)

最大流最小割定理 · qf-mod46-mfmc-equal-value-optimality · candidate





## 81. 最大流最小割定理

完整有限network、全部s–t cuts與forward-edge cut capacity。操作=mfmc-equal-value-optimality；資料={"vertices":["s","a0","a1","a2","t"],"edges":[["s","a0"],["a0","t"],["s","a1"],["a1","t"],["s","a2"],["a2","t"]],"source":"s","sink":"t","capacities":[7,7,8,8,9,9],"flows":[7,7,8,8,9,9]}。請依 [flow_value, minimum_capacity, equal_value_cuts, minimum_cuts, equal_values, maximum_flow_certificate, minimum_cut_certificate, integrality_or_algorithm_claim, certificate] 欄序回答。

同值證書先要求feasible flow；最後否只不提供integrality或特定algorithm證明，不否定maximum/minimum。

查看答案與解釋(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]], 是, 是, 是, 否, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims)

Wave6第九層核對=(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]], 是, 是, 是, 否, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims)；依完整明示資料獨立重算；feasible flow對每個cut皆有value≤capacity；同值flow/cut給雙optimality certificate；單一flow/cut不自動optimal；不計反向edges且不宣稱integrality、augmenting algorithm或一般定理已由instances證明。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r05-max-flow-min-cut-theorem-0081/)

最大流最小割定理 · qf-mod46-mfmc-equal-value-optimality · candidate





## 82. 最大流最小割定理

完整有限network、全部s–t cuts與forward-edge cut capacity。操作=mfmc-equal-value-optimality；資料={"vertices":["s","a0","a1","a2","a3","t"],"edges":[["s","a0"],["a0","t"],["s","a1"],["a1","t"],["s","a2"],["a2","t"],["s","a3"],["a3","t"],["a0","s"]],"source":"s","sink":"t","capacities":[8,8,9,9,10,10,11,11,13],"flows":[8,8,9,9,10,10,11,11,0]}。請依 [flow_value, minimum_capacity, equal_value_cuts, minimum_cuts, equal_values, maximum_flow_certificate, minimum_cut_certificate, integrality_or_algorithm_claim, certificate] 欄序回答。

同值證書先要求feasible flow；最後否只不提供integrality或特定algorithm證明，不否定maximum/minimum。

查看答案與解釋(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]], 是, 是, 是, 否, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims)

Wave6第九層核對=(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]], 是, 是, 是, 否, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims)；依完整明示資料獨立重算；feasible flow對每個cut皆有value≤capacity；同值flow/cut給雙optimality certificate；單一flow/cut不自動optimal；不計反向edges且不宣稱integrality、augmenting algorithm或一般定理已由instances證明。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r05-max-flow-min-cut-theorem-0082/)

最大流最小割定理 · qf-mod46-mfmc-equal-value-optimality · candidate





## 83. 最大流最小割定理

完整有限network、全部s–t cuts與forward-edge cut capacity。操作=mfmc-equal-value-optimality；資料={"vertices":["s","a0","a1","a2","a3","a4","t"],"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","capacities":[9,9,10,10,11,11,12,12,13,13],"flows":[9,9,10,10,11,11,12,12,13,13]}。請依 [flow_value, minimum_capacity, equal_value_cuts, minimum_cuts, equal_values, maximum_flow_certificate, minimum_cut_certificate, integrality_or_algorithm_claim, certificate] 欄序回答。

同值證書先要求feasible flow；最後否只不提供integrality或特定algorithm證明，不否定maximum/minimum。

查看答案與解釋(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]], 是, 是, 是, 否, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims)

Wave6第九層核對=(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]], 是, 是, 是, 否, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims)；依完整明示資料獨立重算；feasible flow對每個cut皆有value≤capacity；同值flow/cut給雙optimality certificate；單一flow/cut不自動optimal；不計反向edges且不宣稱integrality、augmenting algorithm或一般定理已由instances證明。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r05-max-flow-min-cut-theorem-0083/)

最大流最小割定理 · qf-mod46-mfmc-equal-value-optimality · candidate





## 84. 最大流最小割定理

完整有限network、全部s–t cuts與forward-edge cut capacity。操作=mfmc-equal-value-optimality；資料={"vertices":["s","a0","a1","t"],"edges":[["s","a0"],["a0","t"],["s","a1"],["a1","t"]],"source":"s","sink":"t","capacities":[10,10,11,11],"flows":[10,10,11,11]}。請依 [flow_value, minimum_capacity, equal_value_cuts, minimum_cuts, equal_values, maximum_flow_certificate, minimum_cut_certificate, integrality_or_algorithm_claim, certificate] 欄序回答。

同值證書先要求feasible flow；最後否只不提供integrality或特定algorithm證明，不否定maximum/minimum。

查看答案與解釋(21, 21, [[s], [a0, s], [a1, s], [a0, a1, s]], [[s], [a0, s], [a1, s], [a0, a1, s]], 是, 是, 是, 否, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims)

Wave6第九層核對=(21, 21, [[s], [a0, s], [a1, s], [a0, a1, s]], [[s], [a0, s], [a1, s], [a0, a1, s]], 是, 是, 是, 否, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims)；依完整明示資料獨立重算；feasible flow對每個cut皆有value≤capacity；同值flow/cut給雙optimality certificate；單一flow/cut不自動optimal；不計反向edges且不宣稱integrality、augmenting algorithm或一般定理已由instances證明。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r05-max-flow-min-cut-theorem-0084/)

最大流最小割定理 · qf-mod46-mfmc-equal-value-optimality · candidate





## 85. 最大流最小割定理

完整有限network、全部s–t cuts與forward-edge cut capacity。操作=mfmc-equal-value-optimality；資料={"vertices":["s","a0","a1","a2","t"],"edges":[["s","a0"],["a0","t"],["s","a1"],["a1","t"],["s","a2"],["a2","t"],["a0","s"]],"source":"s","sink":"t","capacities":[11,11,12,12,13,13,16],"flows":[11,11,12,12,13,13,0]}。請依 [flow_value, minimum_capacity, equal_value_cuts, minimum_cuts, equal_values, maximum_flow_certificate, minimum_cut_certificate, integrality_or_algorithm_claim, certificate] 欄序回答。

同值證書先要求feasible flow；最後否只不提供integrality或特定algorithm證明，不否定maximum/minimum。

查看答案與解釋(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]], 是, 是, 是, 否, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims)

Wave6第九層核對=(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]], 是, 是, 是, 否, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims)；依完整明示資料獨立重算；feasible flow對每個cut皆有value≤capacity；同值flow/cut給雙optimality certificate；單一flow/cut不自動optimal；不計反向edges且不宣稱integrality、augmenting algorithm或一般定理已由instances證明。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r05-max-flow-min-cut-theorem-0085/)

最大流最小割定理 · qf-mod46-mfmc-equal-value-optimality · candidate





## 86. 最大流最小割定理

完整有限network、全部s–t cuts與forward-edge cut capacity。操作=mfmc-equal-value-optimality；資料={"vertices":["s","a0","a1","a2","a3","t"],"edges":[["s","a0"],["a0","t"],["s","a1"],["a1","t"],["s","a2"],["a2","t"],["s","a3"],["a3","t"]],"source":"s","sink":"t","capacities":[12,12,13,13,14,14,15,15],"flows":[12,12,13,13,14,14,15,15]}。請依 [flow_value, minimum_capacity, equal_value_cuts, minimum_cuts, equal_values, maximum_flow_certificate, minimum_cut_certificate, integrality_or_algorithm_claim, certificate] 欄序回答。

同值證書先要求feasible flow；最後否只不提供integrality或特定algorithm證明，不否定maximum/minimum。

查看答案與解釋(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]], 是, 是, 是, 否, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims)

Wave6第九層核對=(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]], 是, 是, 是, 否, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims)；依完整明示資料獨立重算；feasible flow對每個cut皆有value≤capacity；同值flow/cut給雙optimality certificate；單一flow/cut不自動optimal；不計反向edges且不宣稱integrality、augmenting algorithm或一般定理已由instances證明。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r05-max-flow-min-cut-theorem-0086/)

最大流最小割定理 · qf-mod46-mfmc-equal-value-optimality · candidate





## 87. 最大流最小割定理

完整有限network、全部s–t cuts與forward-edge cut capacity。操作=mfmc-equal-value-optimality；資料={"vertices":["s","a0","a1","a2","a3","a4","t"],"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","capacities":[13,13,14,14,15,15,16,16,17,17],"flows":[13,13,14,14,15,15,16,16,17,17]}。請依 [flow_value, minimum_capacity, equal_value_cuts, minimum_cuts, equal_values, maximum_flow_certificate, minimum_cut_certificate, integrality_or_algorithm_claim, certificate] 欄序回答。

同值證書先要求feasible flow；最後否只不提供integrality或特定algorithm證明，不否定maximum/minimum。

查看答案與解釋(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]], 是, 是, 是, 否, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims)

Wave6第九層核對=(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]], 是, 是, 是, 否, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims)；依完整明示資料獨立重算；feasible flow對每個cut皆有value≤capacity；同值flow/cut給雙optimality certificate；單一flow/cut不自動optimal；不計反向edges且不宣稱integrality、augmenting algorithm或一般定理已由instances證明。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r05-max-flow-min-cut-theorem-0087/)

最大流最小割定理 · qf-mod46-mfmc-equal-value-optimality · candidate





## 88. 最大流最小割定理

完整有限network、全部s–t cuts與forward-edge cut capacity。操作=mfmc-equal-value-optimality；資料={"vertices":["s","a0","a1","t"],"edges":[["s","a0"],["a0","t"],["s","a1"],["a1","t"],["a0","s"]],"source":"s","sink":"t","capacities":[14,14,15,15,19],"flows":[14,14,15,15,0]}。請依 [flow_value, minimum_capacity, equal_value_cuts, minimum_cuts, equal_values, maximum_flow_certificate, minimum_cut_certificate, integrality_or_algorithm_claim, certificate] 欄序回答。

同值證書先要求feasible flow；最後否只不提供integrality或特定algorithm證明，不否定maximum/minimum。

查看答案與解釋(29, 29, [[s], [a0, s], [a1, s], [a0, a1, s]], [[s], [a0, s], [a1, s], [a0, a1, s]], 是, 是, 是, 否, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims)

Wave6第九層核對=(29, 29, [[s], [a0, s], [a1, s], [a0, a1, s]], [[s], [a0, s], [a1, s], [a0, a1, s]], 是, 是, 是, 否, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims)；依完整明示資料獨立重算；feasible flow對每個cut皆有value≤capacity；同值flow/cut給雙optimality certificate；單一flow/cut不自動optimal；不計反向edges且不宣稱integrality、augmenting algorithm或一般定理已由instances證明。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r05-max-flow-min-cut-theorem-0088/)

最大流最小割定理 · qf-mod46-mfmc-equal-value-optimality · candidate





## 89. 最大流最小割定理

完整有限network、全部s–t cuts與forward-edge cut capacity。操作=mfmc-equal-value-optimality；資料={"vertices":["s","a0","a1","a2","t"],"edges":[["s","a0"],["a0","t"],["s","a1"],["a1","t"],["s","a2"],["a2","t"]],"source":"s","sink":"t","capacities":[15,15,16,16,17,17],"flows":[15,15,16,16,17,17]}。請依 [flow_value, minimum_capacity, equal_value_cuts, minimum_cuts, equal_values, maximum_flow_certificate, minimum_cut_certificate, integrality_or_algorithm_claim, certificate] 欄序回答。

同值證書先要求feasible flow；最後否只不提供integrality或特定algorithm證明，不否定maximum/minimum。

查看答案與解釋(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]], 是, 是, 是, 否, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims)

Wave6第九層核對=(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]], 是, 是, 是, 否, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims)；依完整明示資料獨立重算；feasible flow對每個cut皆有value≤capacity；同值flow/cut給雙optimality certificate；單一flow/cut不自動optimal；不計反向edges且不宣稱integrality、augmenting algorithm或一般定理已由instances證明。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r05-max-flow-min-cut-theorem-0089/)

最大流最小割定理 · qf-mod46-mfmc-equal-value-optimality · candidate





## 90. 最大流最小割定理

完整有限network、全部s–t cuts與forward-edge cut capacity。操作=mfmc-equal-value-optimality；資料={"vertices":["s","a0","a1","a2","a3","t"],"edges":[["s","a0"],["a0","t"],["s","a1"],["a1","t"],["s","a2"],["a2","t"],["s","a3"],["a3","t"]],"source":"s","sink":"t","capacities":[16,16,17,17,18,18,19,19],"flows":[16,16,17,17,18,18,19,19]}。請依 [flow_value, minimum_capacity, equal_value_cuts, minimum_cuts, equal_values, maximum_flow_certificate, minimum_cut_certificate, integrality_or_algorithm_claim, certificate] 欄序回答。

同值證書先要求feasible flow；最後否只不提供integrality或特定algorithm證明，不否定maximum/minimum。

查看答案與解釋(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]], 是, 是, 是, 否, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims)

Wave6第九層核對=(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]], 是, 是, 是, 否, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims)；依完整明示資料獨立重算；feasible flow對每個cut皆有value≤capacity；同值flow/cut給雙optimality certificate；單一flow/cut不自動optimal；不計反向edges且不宣稱integrality、augmenting algorithm或一般定理已由instances證明。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r05-max-flow-min-cut-theorem-0090/)

最大流最小割定理 · qf-mod46-mfmc-equal-value-optimality · candidate





## 91. 最大流最小割定理

完整有限network、全部s–t cuts與forward-edge cut capacity。操作=mfmc-equal-value-optimality；資料={"vertices":["s","a0","a1","a2","a3","a4","t"],"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","capacities":[17,17,18,18,19,19,20,20,21,21,22],"flows":[17,17,18,18,19,19,20,20,21,21,0]}。請依 [flow_value, minimum_capacity, equal_value_cuts, minimum_cuts, equal_values, maximum_flow_certificate, minimum_cut_certificate, integrality_or_algorithm_claim, certificate] 欄序回答。

同值證書先要求feasible flow；最後否只不提供integrality或特定algorithm證明，不否定maximum/minimum。

查看答案與解釋(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]], 是, 是, 是, 否, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims)

Wave6第九層核對=(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]], 是, 是, 是, 否, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims)；依完整明示資料獨立重算；feasible flow對每個cut皆有value≤capacity；同值flow/cut給雙optimality certificate；單一flow/cut不自動optimal；不計反向edges且不宣稱integrality、augmenting algorithm或一般定理已由instances證明。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r05-max-flow-min-cut-theorem-0091/)

最大流最小割定理 · qf-mod46-mfmc-equal-value-optimality · candidate





## 92. 最大流最小割定理

完整有限network、全部s–t cuts與forward-edge cut capacity。操作=mfmc-equal-value-optimality；資料={"vertices":["s","a0","a1","t"],"edges":[["s","a0"],["a0","t"],["s","a1"],["a1","t"]],"source":"s","sink":"t","capacities":[18,18,19,19],"flows":[18,18,19,19]}。請依 [flow_value, minimum_capacity, equal_value_cuts, minimum_cuts, equal_values, maximum_flow_certificate, minimum_cut_certificate, integrality_or_algorithm_claim, certificate] 欄序回答。

同值證書先要求feasible flow；最後否只不提供integrality或特定algorithm證明，不否定maximum/minimum。

查看答案與解釋(37, 37, [[s], [a0, s], [a1, s], [a0, a1, s]], [[s], [a0, s], [a1, s], [a0, a1, s]], 是, 是, 是, 否, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims)

Wave6第九層核對=(37, 37, [[s], [a0, s], [a1, s], [a0, a1, s]], [[s], [a0, s], [a1, s], [a0, a1, s]], 是, 是, 是, 否, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims)；依完整明示資料獨立重算；feasible flow對每個cut皆有value≤capacity；同值flow/cut給雙optimality certificate；單一flow/cut不自動optimal；不計反向edges且不宣稱integrality、augmenting algorithm或一般定理已由instances證明。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r05-max-flow-min-cut-theorem-0092/)

最大流最小割定理 · qf-mod46-mfmc-equal-value-optimality · candidate





## 93. 最大流最小割定理

完整有限network、全部s–t cuts與forward-edge cut capacity。操作=mfmc-equal-value-optimality；資料={"vertices":["s","a0","a1","a2","t"],"edges":[["s","a0"],["a0","t"],["s","a1"],["a1","t"],["s","a2"],["a2","t"]],"source":"s","sink":"t","capacities":[19,19,20,20,21,21],"flows":[19,19,20,20,21,21]}。請依 [flow_value, minimum_capacity, equal_value_cuts, minimum_cuts, equal_values, maximum_flow_certificate, minimum_cut_certificate, integrality_or_algorithm_claim, certificate] 欄序回答。

同值證書先要求feasible flow；最後否只不提供integrality或特定algorithm證明，不否定maximum/minimum。

查看答案與解釋(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]], 是, 是, 是, 否, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims)

Wave6第九層核對=(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]], 是, 是, 是, 否, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims)；依完整明示資料獨立重算；feasible flow對每個cut皆有value≤capacity；同值flow/cut給雙optimality certificate；單一flow/cut不自動optimal；不計反向edges且不宣稱integrality、augmenting algorithm或一般定理已由instances證明。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r05-max-flow-min-cut-theorem-0093/)

最大流最小割定理 · qf-mod46-mfmc-equal-value-optimality · candidate





## 94. 最大流最小割定理

完整有限network、全部s–t cuts與forward-edge cut capacity。操作=mfmc-equal-value-optimality；資料={"vertices":["s","a0","a1","a2","a3","t"],"edges":[["s","a0"],["a0","t"],["s","a1"],["a1","t"],["s","a2"],["a2","t"],["s","a3"],["a3","t"],["a0","s"]],"source":"s","sink":"t","capacities":[20,20,21,21,22,22,23,23,25],"flows":[20,20,21,21,22,22,23,23,0]}。請依 [flow_value, minimum_capacity, equal_value_cuts, minimum_cuts, equal_values, maximum_flow_certificate, minimum_cut_certificate, integrality_or_algorithm_claim, certificate] 欄序回答。

同值證書先要求feasible flow；最後否只不提供integrality或特定algorithm證明，不否定maximum/minimum。

查看答案與解釋(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]], 是, 是, 是, 否, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims)

Wave6第九層核對=(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]], 是, 是, 是, 否, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims)；依完整明示資料獨立重算；feasible flow對每個cut皆有value≤capacity；同值flow/cut給雙optimality certificate；單一flow/cut不自動optimal；不計反向edges且不宣稱integrality、augmenting algorithm或一般定理已由instances證明。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r05-max-flow-min-cut-theorem-0094/)

最大流最小割定理 · qf-mod46-mfmc-equal-value-optimality · candidate





## 95. 最大流最小割定理

完整有限network、全部s–t cuts與forward-edge cut capacity。操作=mfmc-equal-value-optimality；資料={"vertices":["s","a0","a1","a2","a3","a4","t"],"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","capacities":[21,21,22,22,23,23,24,24,25,25],"flows":[21,21,22,22,23,23,24,24,25,25]}。請依 [flow_value, minimum_capacity, equal_value_cuts, minimum_cuts, equal_values, maximum_flow_certificate, minimum_cut_certificate, integrality_or_algorithm_claim, certificate] 欄序回答。

同值證書先要求feasible flow；最後否只不提供integrality或特定algorithm證明，不否定maximum/minimum。

查看答案與解釋(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]], 是, 是, 是, 否, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims)

Wave6第九層核對=(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]], 是, 是, 是, 否, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims)；依完整明示資料獨立重算；feasible flow對每個cut皆有value≤capacity；同值flow/cut給雙optimality certificate；單一flow/cut不自動optimal；不計反向edges且不宣稱integrality、augmenting algorithm或一般定理已由instances證明。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r05-max-flow-min-cut-theorem-0095/)

最大流最小割定理 · qf-mod46-mfmc-equal-value-optimality · candidate





## 96. 最大流最小割定理

完整有限network、全部s–t cuts與forward-edge cut capacity。操作=mfmc-equal-value-optimality；資料={"vertices":["s","a0","a1","t"],"edges":[["s","a0"],["a0","t"],["s","a1"],["a1","t"]],"source":"s","sink":"t","capacities":[22,22,23,23],"flows":[22,22,23,23]}。請依 [flow_value, minimum_capacity, equal_value_cuts, minimum_cuts, equal_values, maximum_flow_certificate, minimum_cut_certificate, integrality_or_algorithm_claim, certificate] 欄序回答。

同值證書先要求feasible flow；最後否只不提供integrality或特定algorithm證明，不否定maximum/minimum。

查看答案與解釋(45, 45, [[s], [a0, s], [a1, s], [a0, a1, s]], [[s], [a0, s], [a1, s], [a0, a1, s]], 是, 是, 是, 否, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims)

Wave6第九層核對=(45, 45, [[s], [a0, s], [a1, s], [a0, a1, s]], [[s], [a0, s], [a1, s], [a0, a1, s]], 是, 是, 是, 否, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims)；依完整明示資料獨立重算；feasible flow對每個cut皆有value≤capacity；同值flow/cut給雙optimality certificate；單一flow/cut不自動optimal；不計反向edges且不宣稱integrality、augmenting algorithm或一般定理已由instances證明。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r05-max-flow-min-cut-theorem-0096/)

最大流最小割定理 · qf-mod46-mfmc-equal-value-optimality · candidate





## 97. 最大流最小割定理

完整有限network、全部s–t cuts與forward-edge cut capacity。操作=mfmc-equal-value-optimality；資料={"vertices":["s","a0","a1","a2","t"],"edges":[["s","a0"],["a0","t"],["s","a1"],["a1","t"],["s","a2"],["a2","t"],["a0","s"]],"source":"s","sink":"t","capacities":[23,23,24,24,25,25,28],"flows":[23,23,24,24,25,25,0]}。請依 [flow_value, minimum_capacity, equal_value_cuts, minimum_cuts, equal_values, maximum_flow_certificate, minimum_cut_certificate, integrality_or_algorithm_claim, certificate] 欄序回答。

同值證書先要求feasible flow；最後否只不提供integrality或特定algorithm證明，不否定maximum/minimum。

查看答案與解釋(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]], 是, 是, 是, 否, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims)

Wave6第九層核對=(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]], 是, 是, 是, 否, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims)；依完整明示資料獨立重算；feasible flow對每個cut皆有value≤capacity；同值flow/cut給雙optimality certificate；單一flow/cut不自動optimal；不計反向edges且不宣稱integrality、augmenting algorithm或一般定理已由instances證明。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r05-max-flow-min-cut-theorem-0097/)

最大流最小割定理 · qf-mod46-mfmc-equal-value-optimality · candidate





## 98. 最大流最小割定理

完整有限network、全部s–t cuts與forward-edge cut capacity。操作=mfmc-equal-value-optimality；資料={"vertices":["s","a0","a1","a2","a3","t"],"edges":[["s","a0"],["a0","t"],["s","a1"],["a1","t"],["s","a2"],["a2","t"],["s","a3"],["a3","t"]],"source":"s","sink":"t","capacities":[24,24,25,25,26,26,27,27],"flows":[24,24,25,25,26,26,27,27]}。請依 [flow_value, minimum_capacity, equal_value_cuts, minimum_cuts, equal_values, maximum_flow_certificate, minimum_cut_certificate, integrality_or_algorithm_claim, certificate] 欄序回答。

同值證書先要求feasible flow；最後否只不提供integrality或特定algorithm證明，不否定maximum/minimum。

查看答案與解釋(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]], 是, 是, 是, 否, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims)

Wave6第九層核對=(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]], 是, 是, 是, 否, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims)；依完整明示資料獨立重算；feasible flow對每個cut皆有value≤capacity；同值flow/cut給雙optimality certificate；單一flow/cut不自動optimal；不計反向edges且不宣稱integrality、augmenting algorithm或一般定理已由instances證明。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r05-max-flow-min-cut-theorem-0098/)

最大流最小割定理 · qf-mod46-mfmc-equal-value-optimality · candidate





## 99. 最大流最小割定理

完整有限network、全部s–t cuts與forward-edge cut capacity。操作=mfmc-equal-value-optimality；資料={"vertices":["s","a0","a1","a2","a3","a4","t"],"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","capacities":[25,25,26,26,27,27,28,28,29,29],"flows":[25,25,26,26,27,27,28,28,29,29]}。請依 [flow_value, minimum_capacity, equal_value_cuts, minimum_cuts, equal_values, maximum_flow_certificate, minimum_cut_certificate, integrality_or_algorithm_claim, certificate] 欄序回答。

同值證書先要求feasible flow；最後否只不提供integrality或特定algorithm證明，不否定maximum/minimum。

查看答案與解釋(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]], 是, 是, 是, 否, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims)

Wave6第九層核對=(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]], 是, 是, 是, 否, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims)；依完整明示資料獨立重算；feasible flow對每個cut皆有value≤capacity；同值flow/cut給雙optimality certificate；單一flow/cut不自動optimal；不計反向edges且不宣稱integrality、augmenting algorithm或一般定理已由instances證明。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r05-max-flow-min-cut-theorem-0099/)

最大流最小割定理 · qf-mod46-mfmc-equal-value-optimality · candidate





## 100. 最大流最小割定理

完整有限network、全部s–t cuts與forward-edge cut capacity。操作=mfmc-equal-value-optimality；資料={"vertices":["s","a0","a1","t"],"edges":[["s","a0"],["a0","t"],["s","a1"],["a1","t"],["a0","s"]],"source":"s","sink":"t","capacities":[26,26,27,27,31],"flows":[26,26,27,27,0]}。請依 [flow_value, minimum_capacity, equal_value_cuts, minimum_cuts, equal_values, maximum_flow_certificate, minimum_cut_certificate, integrality_or_algorithm_claim, certificate] 欄序回答。

同值證書先要求feasible flow；最後否只不提供integrality或特定algorithm證明，不否定maximum/minimum。

查看答案與解釋(53, 53, [[s], [a0, s], [a1, s], [a0, a1, s]], [[s], [a0, s], [a1, s], [a0, a1, s]], 是, 是, 是, 否, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims)

Wave6第九層核對=(53, 53, [[s], [a0, s], [a1, s], [a0, a1, s]], [[s], [a0, s], [a1, s], [a0, a1, s]], 是, 是, 是, 否, equal_feasible_flow_and_cut_values_certify_both_optimal_without_integrality_or_algorithm_claims)；依完整明示資料獨立重算；feasible flow對每個cut皆有value≤capacity；同值flow/cut給雙optimality certificate；單一flow/cut不自動optimal；不計反向edges且不宣稱integrality、augmenting algorithm或一般定理已由instances證明。



[本題逐步詳解（AI 二次覆核）](https://ocme.evemisslab.com/zh-Hant/questions/q-20261015-advance-20261115-r05-max-flow-min-cut-theorem-0100/)

最大流最小割定理 · qf-mod46-mfmc-equal-value-optimality · candidate



[原始題目 JSONL](https://ocme.evemisslab.com/data/questions/batches/batch-20261015-advance-20261115-r05-max-flow-min-cut-theorem-foundations/questions.jsonl) · [驗收報告](https://ocme.evemisslab.com/data/questions/batches/batch-20261015-advance-20261115-r05-max-flow-min-cut-theorem-foundations/validation-report.json)