sig-procon-regular-02

Transcript

1. ڝϓϩ෼Պձ Regular ୈ 2 ճ Rho17 ͸͡Ίʹ Max Flow-Min Cut Project Selection Problem ڝϓϩ෼Պձ Regular ୈ 2 ճ Rho17 October 9, 2025 Rho17 ڝϓϩ෼Պձ Regular ୈ 2 ճ October 9, 2025 1 / 38
2. ڝϓϩ෼Պձ Regular ୈ 2 ճ Rho17 ͸͡Ίʹ Max Flow-Min Cut Project Selection Problem ໨࣍ 1 ͸͡Ίʹ 2 Max Flow-Min Cut 3 Project Selection Problem Rho17 ڝϓϩ෼Պձ Regular ୈ 2 ճ October 9, 2025 2 / 38
3. ڝϓϩ෼Պձ Regular ୈ 2 ճ Rho17 ͸͡Ίʹ Max Flow-Min Cut Project Selection Problem ໨࣍ 1 ͸͡Ίʹ 2 Max Flow-Min Cut 3 Project Selection Problem Rho17 ڝϓϩ෼Պձ Regular ୈ 2 ճ October 9, 2025 3 / 38
4. ڝϓϩ෼Պձ Regular ୈ 2 ճ Rho17 ͸͡Ίʹ Max Flow-Min Cut Project Selection Problem ࣗݾ঺հ Rho17 • ཧֶ෦৘ใՊֶՊ B4(2025 ೥ ݱࡏ) • AtCoder ԫ (2056) • ෼ՊձॳࢀՃͰ։࠵ଆʹճΔ͜ͱ ʹ...... Rho17 ڝϓϩ෼Պձ Regular ୈ 2 ճ October 9, 2025 4 / 38
5. ڝϓϩ෼Պձ Regular ୈ 2 ճ Rho17 ͸͡Ίʹ Max Flow-Min Cut Project Selection Problem ໨࣍ 1 ͸͡Ίʹ 2 Max Flow-Min Cut 3 Project Selection Problem Rho17 ڝϓϩ෼Պձ Regular ୈ 2 ճ October 9, 2025 5 / 38
6. ڝϓϩ෼Պձ Regular ୈ 2 ճ Rho17 ͸͡Ίʹ Max Flow-Min Cut Project Selection Problem Max Flow ࠷େϑϩʔ N ݸͷڌ఺͕͋Γɺڌ఺ S ͔Βڌ఺ T ·Ͱ෺ࢿΛӡͼ͍ͨɻ෺ࢿ͸ڌ఺ ui ͔ Βڌ఺ vi ʹҰํ௨ߦͰӡͿ͜ͱ͕Ͱ͖ɺ1 ೔͋ͨΓ࠷େͷ༌ૹྔ ci ͕ܾ·ͬͯ ͍Δɻ1 ೔ʹ S ͔Β T ·Ͱ࠷େͰͲΕ͚ͩ༌ૹͰ͖Δ͔ʁ ෺ࢿͷ༌ૹΛςʔϚʹͨ͠ཧ༝͸ޙड़...... Rho17 ڝϓϩ෼Պձ Regular ୈ 2 ճ October 9, 2025 6 / 38
7. ڝϓϩ෼Պձ Regular ୈ 2 ճ Rho17 ͸͡Ίʹ Max Flow-Min Cut Project Selection Problem Max Flow ໰୊Λάϥϑཧ࿦ͷ༻ޠͰݴ͍׵͑Δ ࠷େϑϩʔ N ௖఺ͷ༗޲άϥϑ G = (V, E) ͕͋Γɺ֤ลʹ͸༗ݶͷ༰ྔ c(e) ͕ఆ·ͬͯ ͍Δɻ௖఺ S Λ࢝఺ (Source), ௖఺ T Λऴ఺ (Sink) ͱࢦఆ͢Δɻ ֤ล͔Β࣮਺΁ͷؔ਺ f : E → R ΛҎԼͷ৚݅Λຬͨ͢Α͏ʹఆΊΔɻ • 0 ≤ f(e) ≤ c(e) • ༰ྔ੍໿ • u ∈ V \ {S, T} ʹ͍ͭͯɺ ∑ v∈V f(v, u) = ∑ v∈V f(u, v) • ࢝఺ͱऴ఺Ҏ֎ͰೖΔྔͱग़Δྔ͸ಉ͡ ྲྀྔ |F| = ∑ v∈V f(S, v) Λ maximize ͤΑɻ Rho17 ڝϓϩ෼Պձ Regular ୈ 2 ճ October 9, 2025 7 / 38
8. ڝϓϩ෼Պձ Regular ୈ 2 ճ Rho17 ͸͡Ίʹ Max Flow-Min Cut Project Selection Problem Max Flow ACL ͕ఏڙͯ͘͠Ε͍ͯΔͷͰɺͦΕΛ࢖͏ʢऴʣ ࢀߟɿAtCoder Library ͷυΩϡϝϯτ Listing 1: convolution 1 auto graph=mf_graph<int>(N); 2 for(int i=0;i<E;i++){ 3 graph.add_edge(u[i],v[i],c[i]); 4 } 5 int ans=graph.flow(S,T); Rho17 ڝϓϩ෼Պձ Regular ୈ 2 ճ October 9, 2025 8 / 38
9. ڝϓϩ෼Պձ Regular ୈ 2 ճ Rho17 ͸͡Ίʹ Max Flow-Min Cut Project Selection Problem Max Flow ACL ͕ఏڙͯ͘͠Ε͍ͯΔͷͰɺͦΕΛ࢖͏ʢऴʣ ࢀߟɿAtCoder Library ͷυΩϡϝϯτ Listing 2: convolution 1 auto graph=mf_graph<int>(N); 2 for(int i=0;i<E;i++){ 3 graph.add_edge(u[i],v[i],c[i]); 4 } 5 int ans=graph.flow(S,T); ಺෦ͷΞϧΰϦζϜ͸ʁˠ Ford-Fulkerson ͷΞϧΰϦζϜ 1 1ACL ͷ಺෦͸ Ford-Fulkerson ΑΓߴ଎ͳΞϧΰϦζϜͬΆͦ͞͏ Rho17 ڝϓϩ෼Պձ Regular ୈ 2 ճ October 9, 2025 9 / 38
10. ڝϓϩ෼Պձ Regular ୈ 2 ճ Rho17 ͸͡Ίʹ Max Flow-Min Cut Project Selection Problem Ford-Fulkerson ʮ΋͠ S ͔Β T ΁ͷύε͕࢒͍ͬͯΔͳΒྲྀ͢ʯʁ Rho17 ڝϓϩ෼Պձ Regular ୈ 2 ճ October 9, 2025 10 / 38
11. ڝϓϩ෼Պձ Regular ୈ 2 ճ Rho17 ͸͡Ίʹ Max Flow-Min Cut Project Selection Problem Ford-Fulkerson ʮ΋͠ S ͔Β T ΁ͷύε͕࢒͍ͬͯΔͳΒྲྀ͢ʯʁ Rho17 ڝϓϩ෼Պձ Regular ୈ 2 ճ October 9, 2025 11 / 38
12. ڝϓϩ෼Պձ Regular ୈ 2 ճ Rho17 ͸͡Ίʹ Max Flow-Min Cut Project Selection Problem Ford-Fulkerson ʮ΋͠ S ͔Β T ΁ͷύε͕࢒͍ͬͯΔͳΒྲྀ͢ʯʁ Rho17 ڝϓϩ෼Պձ Regular ୈ 2 ճ October 9, 2025 12 / 38
13. ڝϓϩ෼Պձ Regular ୈ 2 ճ Rho17 ͸͡Ίʹ Max Flow-Min Cut Project Selection Problem Ford-Fulkerson ʮ΋͠ S ͔Β T ΁ͷύε͕࢒͍ͬͯΔͳΒྲྀ͢ʯʁ ࢒͍ͬͯͳ͍ͷͰ͓͠·͍ʁ Rho17 ڝϓϩ෼Պձ Regular ୈ 2 ճ October 9, 2025 13 / 38
14. ڝϓϩ෼Պձ Regular ୈ 2 ճ Rho17 ͸͡Ίʹ Max Flow-Min Cut Project Selection Problem Ford-Fulkerson ʮ΋͠ S ͔Β T ΁ͷύε͕࢒͍ͬͯΔͳΒྲྀ͢ʯʁ ࣮͸ 11 ྲྀͤΔʂˠͲ͏͢Ε͹ 11 ྲྀ͔ͤͨʁ Rho17 ڝϓϩ෼Պձ Regular ୈ 2 ճ October 9, 2025 14 / 38
15. ڝϓϩ෼Պձ Regular ୈ 2 ճ Rho17 ͸͡Ίʹ Max Flow-Min Cut Project Selection Problem Ford-Fulkerson ݱࡏͷϑϩʔΛҰ෦औΓ΍ΊΔૢ࡞͕Ͱ͖Δͱ͍͍͔΋͠Εͳ͍ ࢒༨ωοτϫʔΫ ݱࡏͷϑϩʔ f ʹରͯ͠ɺ cf (e) =              c(e) − f(e) e ∈ E f(e) rev(e) ∈ E 0 otherwise ͱఆٛ͞ΕΔؔ਺ cf Λ࢒༨༰ྔͱ͍͏ɻ௖఺ू߹͕ V Ͱ͋ͬͯล e ͷཁྖ͕ cf (e) Ͱ͋ΔΑ͏ͳάϥϑ Gf Λ࢒༨ωοτϫʔΫͱ͍͏ɻ Rho17 ڝϓϩ෼Պձ Regular ୈ 2 ճ October 9, 2025 15 / 38
16. ڝϓϩ෼Պձ Regular ୈ 2 ճ Rho17 ͸͡Ίʹ Max Flow-Min Cut Project Selection Problem Ford-Fulkerson Gf ্Ͱͷ S ͔Β T ΁ͷύεͰɺ༰ྔͷ࠷খ஋͕ਖ਼ͷ΋ͷͷ͜ͱΛ augmenting path ͱݺͿɻ Augmenting path ͕ݟ͔ͭΕ͹શମͷྲྀྔΛ૿΍ͤΔ (طʹྲྀͨ͠෦෼ΛҰ෦ԡ͠໭ͯ͠ผͷϧʔτʹมߋ͢Δૢ࡞) Rho17 ڝϓϩ෼Պձ Regular ୈ 2 ճ October 9, 2025 16 / 38
17. ڝϓϩ෼Պձ Regular ୈ 2 ճ Rho17 ͸͡Ίʹ Max Flow-Min Cut Project Selection Problem Ford-Fulkerson Augmenting path Λ୳͢ Rho17 ڝϓϩ෼Պձ Regular ୈ 2 ճ October 9, 2025 17 / 38
18. ڝϓϩ෼Պձ Regular ୈ 2 ճ Rho17 ͸͡Ίʹ Max Flow-Min Cut Project Selection Problem Ford-Fulkerson Augmenting path Λ୳͢ Rho17 ڝϓϩ෼Պձ Regular ୈ 2 ճ October 9, 2025 18 / 38
19. ڝϓϩ෼Պձ Regular ୈ 2 ճ Rho17 ͸͡Ίʹ Max Flow-Min Cut Project Selection Problem Ford-Fulkerson Augmenting path Λ୳͢ ͜ΕͰਖ਼͍͠౴͕͑ಘΒΕͨ Rho17 ڝϓϩ෼Պձ Regular ୈ 2 ճ October 9, 2025 19 / 38
20. ڝϓϩ෼Պձ Regular ୈ 2 ճ Rho17 ͸͡Ίʹ Max Flow-Min Cut Project Selection Problem Ford-Fulkerson ͜ΕΛΞϧΰϦζϜʹ͢Ε͹ OK • શͯͷྲྀྔΛ 0 ʹͨ͠ঢ়ଶͰελʔτ • BFS/DFS Ͱ࢒༨άϥϑ্ͷ S − T ύεΛݟ͚ͭΔ • ύε͕ݟ͔ͭΕ͹ύε্ͷ࠷খ஋ͱಉ͚ͩ͡ྲྀ͠ɺ࢒༨άϥϑΛߋ৽ • ύε͕ݟ͔ͭΒͳ͚Ε͹͓͠·͍ ܭࢉྔ͸ʁ F ͕ 1 ͣͭ૿͑ΔΑ͏ͳύλʔϯ͕࠷ѱͳͷͰ O(FE) (ݱ࣮తʹ 1 ͣͭ૿͑Δ͜ͱ͸͋·Γͳ͘ɺߴ଎ʹಈ͘͜ͱ͕ଟ͍) Rho17 ڝϓϩ෼Պձ Regular ୈ 2 ճ October 9, 2025 20 / 38
21. ڝϓϩ෼Պձ Regular ୈ 2 ճ Rho17 ͸͡Ίʹ Max Flow-Min Cut Project Selection Problem Edmonds-Karp/Dinic S ͔Βͷڑ཭Λ BFS Ͱௐ΂ΔΑ͏ʹ͠ɺ • S ͔Βͷڑ཭Λ BFS Ͱௐ΂Δ • DFS Ͱ BFS Tree ্ͷ࠷୹ύεΛऔಘͰ͖Δ͔͗Γऔಘ͢Δ • ࢒༨άϥϑΛߋ৽͠ɺBFS ʹ໭Δ ͜ͷΑ͏ʹ͢Δͱܭࢉྔ͕ O(V 2E)ɺແବͳ୳ࡧΛ͠ͳ͍Α͏ʹ T ͔Β΋ BFS ͯ͠౸ୡՄೳੑΛௐ΂ΔͳͲ͢Δͱ O(V E2) ʹͳΔʢEdmonds-Karp/Dinic ๏ʣ Rho17 ڝϓϩ෼Պձ Regular ୈ 2 ճ October 9, 2025 21 / 38
22. ڝϓϩ෼Պձ Regular ୈ 2 ճ Rho17 ͸͡Ίʹ Max Flow-Min Cut Project Selection Problem Edmonds-Karp/Dinic ূ໌ͷུ֓ • 1 ճ BFS Λߦ͏ͨͼʹ S → T ͷ࠷୹ augmenting path ͷ௕͕͞ 1 ͣͭ૿ ͑ΔͷͰɺBFS ͷճ਺͕ O(V ) • ֤ DFS ͸ O(E) Ͱɺ͜ΕΛ࠷େ O(E) ճߦ͏ ࢒༨άϥϑΛߋ৽͢ΔλΠϛϯάͰ͸࠷୹ڑ཭͕ඞͣ૿Ճ͢Δ͜ͱΛؤுͬͯ ݴ͑͹Α͍ Rho17 ڝϓϩ෼Պձ Regular ୈ 2 ճ October 9, 2025 22 / 38
23. ڝϓϩ෼Պձ Regular ୈ 2 ճ Rho17 ͸͡Ίʹ Max Flow-Min Cut Project Selection Problem Min Cut ࠷খΧοτ N ݸͷڌ఺͕͋Γɺڌ఺ S ͔Βڌ఺ T ·Ͱ෺ࢿ͕ӡ͹Ε͍ͯΔɻڌ఺ ui ͔Β ڌ఺ vi ͷ޲͖ʹ෺ࢿ͕ӡ͹Ε͓ͯΓɺίετ ci Λ෷ͬͯͦͷಓΛഁյ͢Δ͜ ͱ͕Ͱ͖ΔɻS ͔Β T ·Ͱͷύε͕ଘࡏ͠ͳ͘ͳΔͨΊʹඞཁͳίετͷ߹ܭ ͷ࠷খ஋͸ʁ Rho17 ڝϓϩ෼Պձ Regular ୈ 2 ճ October 9, 2025 23 / 38
24. ڝϓϩ෼Պձ Regular ୈ 2 ճ Rho17 ͸͡Ίʹ Max Flow-Min Cut Project Selection Problem Min Cut ޙʑͷͨΊʹݴ͍׵͓͑ͯ͘ ࠷খΧοτ ༗޲άϥϑ G = (V, E) ͱ௖఺ू߹ U(S ∈ U, T / ∈ U) Λߟ͑ΔɻU ͔Β V \ U ʹ޲͔͏ลͷू߹ΛάϥϑͷΧοτͱ͍͏ɻΧοτʹؚ·ΕΔลͷ༰ྔͷ૯࿨ Λ࠷খԽͤΑɻ ͓·͚ɿ Max Cut ͸ NP-Complete ʹͳΔ Rho17 ڝϓϩ෼Պձ Regular ୈ 2 ճ October 9, 2025 24 / 38
25. ڝϓϩ෼Պձ Regular ୈ 2 ճ Rho17 ͸͡Ίʹ Max Flow-Min Cut Project Selection Problem Min Cut Max Flow-Min Cut Equivalence άϥϑ G = (V, E) ͷ࠷େϑϩʔͱ࠷খΧοτͷ஋͸Ұக͢Δɻ ུ֓ɿద౰ͳϑϩʔΛྲྀͨ͠άϥϑͷద౰ͳΧοτΛߟ͑ΔͱɺΧοτͷ༰ྔ ͷํ͕େ͖͍ͷͰʢϑϩʔͷྲྀྔʣ ≤ (Χοτͷ༰ྔ) ͱͳΔɻ ࠷େϑϩʔͷͱ͖ɺ࢒༨ωοτϫʔΫ্Ͱ S → T ͷύε͸ଘࡏ͠ͳ͍ɻͦͷ ͨΊɺ࢒༨ωοτϫʔΫ্Ͱ S ͔Β౸ୡՄೳͳ௖఺Λ U ͱ͢Ε͹ (U, V \ U) ͕ΧοτͱͳΔ͜ͱ͔Β Max Flow ͱ Min Cut ͕౳͘͠ͳΔ Rho17 ڝϓϩ෼Պձ Regular ୈ 2 ճ October 9, 2025 25 / 38
26. ڝϓϩ෼Պձ Regular ୈ 2 ճ Rho17 ͸͡Ίʹ Max Flow-Min Cut Project Selection Problem Min Cut ஫ҙ఺ Ford-Fulkerson ͷΞϧΰϦζϜΑΓ༰ྔ͕੔਺ͷϑϩʔ͸ຖճͷ iteratiron Ͱྲྀ ྔ͕ 1 Ҏ্૿͑Δ͜ͱ͔Βඞͣఀࢭ͢Δɻ ༰ྔ͕ඇ੔਺ͷ৔߹͸ఀࢭ͠ͳ͍৔߹͕͋Δʂ Rho17 ڝϓϩ෼Պձ Regular ୈ 2 ճ October 9, 2025 26 / 38
27. ڝϓϩ෼Պձ Regular ୈ 2 ճ Rho17 ͸͡Ίʹ Max Flow-Min Cut Project Selection Problem ࿅श໰୊ Typical Path Problem (Dif: 2260) ແ޲άϥϑ G = (V, E) ͱ૬ҟͳΔ௖఺ A, B, C ͕༩͑ΒΕΔɻB Λܦ༝ͯ͠ A ͱ C Λ݁Ϳ୯७ύε͕ଘࡏ͢Δ͔൑ఆͤΑɻ • 3 ≤ N ≤ 2 × 105 • N − 1 ≤ M ≤ 2 × 105 ग़య: ABC318-G Rho17 ڝϓϩ෼Պձ Regular ୈ 2 ճ October 9, 2025 27 / 38
28. ڝϓϩ෼Պձ Regular ୈ 2 ճ Rho17 ͸͡Ίʹ Max Flow-Min Cut Project Selection Problem ࿅श໰୊ B → A, B → C ͱ͍͏ 2 ຊͷύεͰ಺఺ૉͳ΋ͷΛͱΕΔ͔ʁ • ʮลૉͳύε 2 ຊʯͳΒ͹ɺS → B, A → T, C → T ͱ௖఺͓ΑͼลΛ௥ Ճ͠ S → T ʹྲྀྔ 2 ͷϑϩʔΛྲྀͤ͹ OK • Menger ͷఆཧɿลૉύεͷ࠷େ਺͸࠷খΧοτͱ౳͍͠ • ಺఺ૉύεͷ࠷େ਺͸࠷খ఺Χοτͱ౳͍͠ • ௖఺ʹؔ͢Δ੍໿ˠ֤௖఺ΛลͷΑ͏ʹѻ͍͍ͨ • ௖఺ u Λ uin, uout ʹ෼ׂɺuin → uout ʹ cap=1 • ล (u, v) ʹର͠ɺuout → vin ʹ cap=1 • S → T ʹྲྀྔ 2 ྲྀΕΕ͹ OK Rho17 ڝϓϩ෼Պձ Regular ୈ 2 ճ October 9, 2025 28 / 38
29. ڝϓϩ෼Պձ Regular ୈ 2 ճ Rho17 ͸͡Ίʹ Max Flow-Min Cut Project Selection Problem ࿅श໰୊ • AtCoder Beginner Contest 205 F - Grid and Tokens (2088) • AtCoder Regular Contest 074 F - Lotus Leaves (2208) • AtCoder Beginner Contest 241 G - Round Robin (2422) • AtCoder Beginner Contest 320 G - Slot Strategy 2 (Hard) (2575) • AtCoder Regular Contest 125 E - Snack (2874) Rho17 ڝϓϩ෼Պձ Regular ୈ 2 ճ October 9, 2025 29 / 38
30. ڝϓϩ෼Պձ Regular ୈ 2 ճ Rho17 ͸͡Ίʹ Max Flow-Min Cut Project Selection Problem ໨࣍ 1 ͸͡Ίʹ 2 Max Flow-Min Cut 3 Project Selection Problem Rho17 ڝϓϩ෼Պձ Regular ୈ 2 ճ October 9, 2025 30 / 38
31. ڝϓϩ෼Պձ Regular ୈ 2 ճ Rho17 ͸͡Ίʹ Max Flow-Min Cut Project Selection Problem Project Selection Problem ೩΍͢ຒΊΔ໰୊ i ൪໨ͷΰϛΛ೩΍ͯ͠ॲཧ͢Δͷʹ͸ Ai ԁ͔͔ΓɺຒΊͯॲཧ͢Δͷʹ͸ Bi ԁ͔͔Δɻ·ͨɺΰϛ i Λ೩΍͠ j ΛຒΊ͍ͯͨ࣌ɺ௥ՃͰ Cij ͷίετ͕͔ ͔ΔɻશͯͷΰϛΛॲཧ͢Δ࠷খίετ͸ʁ ʮ೩΍͢ຒΊΔʯ͸ Project Selection Problem ͷҰ෦ Rho17 ڝϓϩ෼Պձ Regular ୈ 2 ճ October 9, 2025 31 / 38
32. ڝϓϩ෼Պձ Regular ୈ 2 ճ Rho17 ͸͡Ίʹ Max Flow-Min Cut Project Selection Problem Project Selection Problem ֤ΰϛΛʮ೩΍͢ʯଆͱʮຒΊΔʯଆʹॴଐͤ͞Δ໰୊ͩͱࢥ͏ ˠ࠷খΧοτͷܗ=࠷େϑϩʔͰղ͚Δ Rho17 ڝϓϩ෼Պձ Regular ୈ 2 ճ October 9, 2025 32 / 38
33. ڝϓϩ෼Պձ Regular ୈ 2 ճ Rho17 ͸͡Ίʹ Max Flow-Min Cut Project Selection Problem Project Selection Problem Get More Moneyʢ˒ 7ʣ Ո i ʹ͸ݱۚ Ai ԁͱՈ Ci1, · · · , Cij ͷ伴͕ஔ͍ͯ͋ΔɻՈ i ʹೖΔͨΊʹ͸Ո i ͷ伴શ͕ͯඞཁͰɺೖ৔ྉ W ͕͔͔Δɻ ʢೖखֹͨۚ͠ʣ-ʢ෷ͬͨೖ৔ྉʣΛ࠷େԽͤΑɻ ग़య: యܕ 90 ໰- 040 Rho17 ڝϓϩ෼Պձ Regular ୈ 2 ճ October 9, 2025 33 / 38
34. ڝϓϩ෼Պձ Regular ୈ 2 ճ Rho17 ͸͡Ίʹ Max Flow-Min Cut Project Selection Problem Project Selection Problem • ʮX ԁ֫ಘʯͱ͍͏৚݅Λͦͷ··ʮ−X ԁേۚʯͱͯ͠͸͍͚ͳ͍ • ෛลೖΓͷ MaxFlow ͸ NP-Hard • ࠷ॳʹશͯͷ৆ۚ Ai ԁΛճऩ͠ɺ ʮՈ i ʹೖΒͳ͔ͬͨΒ Ai ԁേۚʯͱ ͢Δ • ʮೖΒͳ͍ʯ͔ΒՈʹ༰ྔ WɺՈ͔ΒʮೖΔʯʹ༰ྔ Ai ͷลΛுΔ • Ո i ͷ伴͕Ո j ʹ͋Δͱ͖ɺj ʹೖΒͣͯ͠ i ʹೖΔ͜ͱ͸ແཧ • j ͔Β i ʹ༰ྔ ∞ ͷลΛுΔ Rho17 ڝϓϩ෼Պձ Regular ୈ 2 ճ October 9, 2025 34 / 38
35. ڝϓϩ෼Պձ Regular ୈ 2 ճ Rho17 ͸͡Ίʹ Max Flow-Min Cut Project Selection Problem Project Selection Problem Zebraness (2475) N × N ͷϚε໨͕͋ΓɺനͰృΒΕ͍ͯΔ͔ࠇͰృΒΕ͍ͯΔ͔ܾ·͍ͬͯͳ ͍͔ͷ͍ͣΕ͔Ͱ͋ΔɻҟͳΔ৭ͷϚε͕ลͰ઀͢ΔՕॴΛ࠷େԽͤΑɻ ग़య: ABC193-F Rho17 ڝϓϩ෼Պձ Regular ୈ 2 ճ October 9, 2025 35 / 38
36. ڝϓϩ෼Պձ Regular ୈ 2 ճ Rho17 ͸͡Ίʹ Max Flow-Min Cut Project Selection Problem Project Selection Problem • ଛࣦʹ௚ͯ͠ߟ͑Δ • ʮಉ͡৭͕ྡ઀͢Δʯͱ 1 ԁേۚˠ͜Ε͸ແཧ • Χοτ͸ 2 ͭͷੑ࣭ʹ෼͚Δ໰୊ • άϦου͕༩͑ΒΕΔͷͰɺάϦουͷ 2 ෦άϥϑੑ͔Βࢢদ໛༷ͷยํ ͷ৭Λ൓సͤͯ͞ΈΔ • ʮಉ͡৭͕ྡ઀͢Δʯˠʮҧ͏৭͕ྡ઀͢Δʯ ͱ 1 ԁേۚʹͳΓɺղ͚Δ Α͏ʹͳΔ • ֤ϚεΛ௖఺ʹରԠͤ͞ɺi ͕നͰ j ͕ࠇˠ i → j ʹ༰ྔ 1 ͷล ٯ΋͔͠Γ • ࠷ॳ͔Β֬ఆ͍ͯ͠Δ෦෼ͷѻ͍ʹ஫ҙ Rho17 ڝϓϩ෼Պձ Regular ୈ 2 ճ October 9, 2025 36 / 38
37. ڝϓϩ෼Պձ Regular ୈ 2 ճ Rho17 ͸͡Ίʹ Max Flow-Min Cut Project Selection Problem ࿅श໰୊ • AtCoder Regular Contest 085 E - MUL (2571) • AtCoder Beginner Contest 259 G - Grid Card Game (2428) • AtCoder Beginner Contest 326 G - Unlock Achievement (2470) • AtCoder Beginner Contest 397 G - Maximize Distance (2947) • AtCoder Regular Contest 176 E - Max Vector (3077) • AtCoder Regular Contest 129 E - Yet Another Minimization (3337) Rho17 ڝϓϩ෼Պձ Regular ୈ 2 ճ October 9, 2025 37 / 38
38. ڝϓϩ෼Պձ Regular ୈ 2 ճ Rho17 ͸͡Ίʹ Max Flow-Min Cut Project Selection Problem ࣍ճ༧ࠂ 10 ೔Ͱ͜ͷ෼ྔͷεϥΠυ࡞Δͷ͕͠ΜͲ͍ͷͰ 1 ճ͋ͨΓ΋͏গܰ͘͢͠Δ ͔΋...... ࣍͸࠷খඅ༻ྲྀͱมͳ੍໿ͷѻ͍ํͷ༧ఆͰ͢ Rho17 ڝϓϩ෼Պձ Regular ୈ 2 ճ October 9, 2025 38 / 38