sig_procon_regular_6__1_.pdf

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1. ڝϓϩ෼Պձ Regular ୈ 6 ճ Rho17 ͸͡Ίʹ ҰൠϚονϯά Tutte-Berge ͷఆཧ εϖΫτϧάϥϑ ཧ࿦ ڝϓϩ෼Պձ Regular ୈ 6 ճ Rho17 December 12, 2025 Rho17 ڝϓϩ෼Պձ Regular ୈ 6 ճ December 12, 2025 1 / 25
2. ڝϓϩ෼Պձ Regular ୈ 6 ճ Rho17 ͸͡Ίʹ ҰൠϚονϯά Tutte-Berge ͷఆཧ εϖΫτϧάϥϑ ཧ࿦ ໨࣍ 1 ͸͡Ίʹ 2 ҰൠϚονϯά 3 Tutte-Berge ͷఆཧ 4 εϖΫτϧάϥϑཧ࿦ Rho17 ڝϓϩ෼Պձ Regular ୈ 6 ճ December 12, 2025 2 / 25
3. ڝϓϩ෼Պձ Regular ୈ 6 ճ Rho17 ͸͡Ίʹ ҰൠϚονϯά Tutte-Berge ͷఆཧ εϖΫτϧάϥϑ ཧ࿦ ໨࣍ 1 ͸͡Ίʹ 2 ҰൠϚονϯά 3 Tutte-Berge ͷఆཧ 4 εϖΫτϧάϥϑཧ࿦ Rho17 ڝϓϩ෼Պձ Regular ୈ 6 ճ December 12, 2025 3 / 25
4. ڝϓϩ෼Պձ Regular ୈ 6 ճ Rho17 ͸͡Ίʹ ҰൠϚονϯά Tutte-Berge ͷఆཧ εϖΫτϧάϥϑ ཧ࿦ ࣗݾ঺հ Rho17 • ཧֶ෦৘ใՊֶՊ B4(2025 ೥ ݱࡏ) • AtCoder ੨ (1922) • ͋Γ͑Μྫྷ͑ํͯ͠ྦ Rho17 ڝϓϩ෼Պձ Regular ୈ 6 ճ December 12, 2025 4 / 25
5. ڝϓϩ෼Պձ Regular ୈ 6 ճ Rho17 ͸͡Ίʹ ҰൠϚονϯά Tutte-Berge ͷఆཧ εϖΫτϧάϥϑ ཧ࿦ ໨࣍ 1 ͸͡Ίʹ 2 ҰൠϚονϯά 3 Tutte-Berge ͷఆཧ 4 εϖΫτϧάϥϑཧ࿦ Rho17 ڝϓϩ෼Պձ Regular ୈ 6 ճ December 12, 2025 5 / 25
6. ڝϓϩ෼Պձ Regular ୈ 6 ճ Rho17 ͸͡Ίʹ ҰൠϚονϯά Tutte-Berge ͷఆཧ εϖΫτϧάϥϑ ཧ࿦ ҰൠϚονϯά ҰൠϚονϯά ೋ෦ͱ͸ݶΒͳ͍άϥϑ G(V, E) ͕͋ΔɻબΜͩͲͷ 2 ล΋୺఺Λڞ༗͠ͳ͍ Α͏ʹลΛ͍͔ͭ͘બͿɻ࠷େͰԿຊબ΂Δʁ ͜ͷΑ͏ʹ୺఺Λڞ༗͠ͳ͍ลͷ෦෼ू߹ M ⊆ E ΛϚονϯάͱ͍͏ɽ Rho17 ڝϓϩ෼Պձ Regular ୈ 6 ճ December 12, 2025 6 / 25
7. ڝϓϩ෼Պձ Regular ୈ 6 ճ Rho17 ͸͡Ίʹ ҰൠϚονϯά Tutte-Berge ͷఆཧ εϖΫτϧάϥϑ ཧ࿦ ҰൠϚονϯά ΋͸΍ೋ෦άϥϑͰ͸ͳ͍ͷͰϑϩʔͳͲͰղ͘͜ͱ͸Ͱ͖ͳ͍ɻ ҰൠϚονϯάࣗମ͸ଟ߲ࣜ࣌ؒͰղ͚Δɻ • O(V 3) : Edmonds ͷՖ෼ղ • O(E √ V )ɿMicali&Vazirani(1980) ݱࡏ࠷଎ ࣮૷͕͋Δ͔͸ෆ໌ ͍ͣΕ΋࣮૷͕Ϡό͗͢ΔͷͰڝϓϩͰཁٻ͞ΕΔ͜ͱ͸ͳ͍͸ͣ Rho17 ڝϓϩ෼Պձ Regular ୈ 6 ճ December 12, 2025 7 / 25
8. ڝϓϩ෼Պձ Regular ୈ 6 ճ Rho17 ͸͡Ίʹ ҰൠϚονϯά Tutte-Berge ͷఆཧ εϖΫτϧάϥϑ ཧ࿦ ҰൠϚονϯά ೋ෦Ϛονϯάͷࡍͷ Augmenting Path ͷٞ࿦͸࢖͑Δɻ Augmenting Path Ϛονϯάʹ࢖ΘΕ͍ͯͳ͍ล͔Β։࢝ͯ͠࢖ΘΕ͍ͯͳ͍ลͱ࢖ΘΕ͍ͯΔ ลΛަޓʹͨͲΓɺ࢖ΘΕ͍ͯͳ͍ลͰऴྃ͢ΔΑ͏ͳύεΛ Augmenting Path ͱݺͿɻ࠷େϚονϯάͱ Augmenting Path ͕ଘࡏ͠ͳ͍͜ͱ͸ಉ஋Ͱ ͋Δɻ Rho17 ڝϓϩ෼Պձ Regular ୈ 6 ճ December 12, 2025 8 / 25
9. ڝϓϩ෼Պձ Regular ୈ 6 ճ Rho17 ͸͡Ίʹ ҰൠϚονϯά Tutte-Berge ͷఆཧ εϖΫτϧάϥϑ ཧ࿦ ҰൠϚονϯά ΋͠ح਺௕ (2k + 1) ͷด࿏Ͱ͋ͬͯϚονϯάʹ k ຊ࢖͍ͬͯΔΑ͏ͳ΋ͷ ʢBlossomʣ͕͋Ε͹ɺͦͷด࿏Λ 1 ఺ʹॖ໿͢Δɻ Figure: Blossom Rho17 ڝϓϩ෼Պձ Regular ୈ 6 ճ December 12, 2025 9 / 25
10. ڝϓϩ෼Պձ Regular ୈ 6 ճ Rho17 ͸͡Ίʹ ҰൠϚονϯά Tutte-Berge ͷఆཧ εϖΫτϧάϥϑ ཧ࿦ ҰൠϚονϯά Blossom Λॖ໿͢Δ͜ͱʹΑͬͯ࠷େϚονϯά͔Ͳ͏͔͸มԽ͠ͳ͍ͨΊɺ Augmenting Path Λ୳ࡧ͠ Blossom ͕Ͱ͖࣍ୈॖ໿ͯ͠௵͢͜ͱΛ܁Γฦͤ͹ Α͍ɻ O(V 4) ͷ͸ͣ ࣮ࡍ͸΋͏গ͠޻෉͢Δͱܭࢉྔ͕ O(V 3) ʹͳΔɻ Rho17 ڝϓϩ෼Պձ Regular ୈ 6 ճ December 12, 2025 10 / 25
11. ڝϓϩ෼Պձ Regular ୈ 6 ճ Rho17 ͸͡Ίʹ ҰൠϚονϯά Tutte-Berge ͷఆཧ εϖΫτϧάϥϑ ཧ࿦ ໨࣍ 1 ͸͡Ίʹ 2 ҰൠϚονϯά 3 Tutte-Berge ͷఆཧ 4 εϖΫτϧάϥϑཧ࿦ Rho17 ڝϓϩ෼Պձ Regular ୈ 6 ճ December 12, 2025 11 / 25
12. ڝϓϩ෼Պձ Regular ୈ 6 ճ Rho17 ͸͡Ίʹ ҰൠϚονϯά Tutte-Berge ͷఆཧ εϖΫτϧάϥϑ ཧ࿦ Tutte-Berge ͷఆཧ Tutte-Berge ͷఆཧ ແ޲άϥϑ G(V, E) ʹର͠ɼ࠷େϚονϯάͷαΠζ͸ min S⊆V (|S| − q(G − S) + |V |)/2 Ͱ༩͑ΒΕΔɽ͜͜Ͱ q(U) ͸෦෼άϥϑ U ͷதͰ࿈݁੒෼ͷαΠζ͕ح਺Ͱ ͋Δ΋ͷͷݸ਺ɽ Rho17 ڝϓϩ෼Պձ Regular ୈ 6 ճ December 12, 2025 12 / 25
13. ڝϓϩ෼Պձ Regular ୈ 6 ճ Rho17 ͸͡Ίʹ ҰൠϚονϯά Tutte-Berge ͷఆཧ εϖΫτϧάϥϑ ཧ࿦ Tutte-Berge ͷఆཧ ূ໌ʹ͸࣍ͷิ୊Λ༻͍Δɽ Tutte’s 1-factor theorem άϥϑ͕׬શϚονϯάΛ࣋ͭ͜ͱ͸ q(G − S) ≤ |S| ͕೚ҙͷ S ⊆ V (G) ʹͭ ͍ͯ੒Γཱͭ͜ͱͱಉ஋Ͱ͋Δɽ ⇒ ํ޲͸؆୯ɽOdd Components ͕ 1 ͭͰ͖Δͷʹର͠ S ͷͲΕ͔ͱ݁Ϳඞཁ ͕͋Δɽ ⇐ ͸େมɽ௖఺਺ʹؔ͢Δؼೲ๏Ͱূ໌͢Δɽ Rho17 ڝϓϩ෼Պձ Regular ୈ 6 ճ December 12, 2025 13 / 25
14. ڝϓϩ෼Պձ Regular ୈ 6 ճ Rho17 ͸͡Ίʹ ҰൠϚονϯά Tutte-Berge ͷఆཧ εϖΫτϧάϥϑ ཧ࿦ Tutte-Berge ͷఆཧ ʢূ໌ͷུ֓ʣTutte condition ͕੒Γཱ͍ͬͯΔͱԾఆ͢ΔɽV (G) ͷ෦෼ू ߹Ͱ͋ͬͯɼq(G − S0) = |S0| Λຬͨ͢ۃେͳ΋ͷΛ 1 ͭͱΔɽG − S0 ͷ odd components ͱ even components ΛͦΕͧΕ Q1, · · · , Qm, D1, · · · , Dk ͱ͢Δɽ ͜ͷͱ͖ɼҎԼ͕ݴ͑Δɽ ʢৄࡉུʣ 1 Dj ͸׬શϚονϯάΛ࣋ͭɽ 2 v ∈ Qi ͷऔΓํʹΑΒͣ Qi − v ͸׬શϚονϯάΛ࣋ͭɽ 3 Q ͷ௖఺ͱ S0 ͷؒͰ׬શϚονϯάΛ૊Ή͜ͱ͕Ͱ͖Δɽ Rho17 ڝϓϩ෼Պձ Regular ୈ 6 ճ December 12, 2025 14 / 25
15. ڝϓϩ෼Պձ Regular ୈ 6 ճ Rho17 ͸͡Ίʹ ҰൠϚονϯά Tutte-Berge ͷఆཧ εϖΫτϧάϥϑ ཧ࿦ Tutte-Berge ͷఆཧ I Λ௖఺਺ |V | ͱۮح͕౳͍͠੔਺ͱ͢ΔɽG ʹ௖఺਺ I ͷ׬શάϥϑΛՃ ͑ɼ׬શάϥϑͱݩͷάϥϑͷؒશͯʹลΛுͬͨ΋ͷΛ G′ ͱ͢ΔɽG ͕α Πζ (V − I)/2 ͷϚονϯάΛ࣋ͭ͜ͱͱ G′ ͕׬શϚονϯάΛ࣋ͭ͜ͱ͸ ಉ஋ͳͷͰɼG′ ʹ Tutte’s 1-factor theorem Λద༻͢ΔɽಘΒΕͨ I ͷ࠷খ஋ ͕Ϛονϯάʹ࢖Θͳ͍௖఺ͷ࠷খ஋ɽ Rho17 ڝϓϩ෼Պձ Regular ୈ 6 ճ December 12, 2025 15 / 25
16. ڝϓϩ෼Պձ Regular ୈ 6 ճ Rho17 ͸͡Ίʹ ҰൠϚονϯά Tutte-Berge ͷఆཧ εϖΫτϧάϥϑ ཧ࿦ ໨࣍ 1 ͸͡Ίʹ 2 ҰൠϚονϯά 3 Tutte-Berge ͷఆཧ 4 εϖΫτϧάϥϑཧ࿦ Rho17 ڝϓϩ෼Պձ Regular ୈ 6 ճ December 12, 2025 16 / 25
17. ڝϓϩ෼Պձ Regular ୈ 6 ճ Rho17 ͸͡Ίʹ ҰൠϚονϯά Tutte-Berge ͷఆཧ εϖΫτϧάϥϑ ཧ࿦ εϖΫτϧάϥϑཧ࿦ εϖΫτϧάϥϑཧ࿦ͱ͸ʁˠઢܗ୅਺ͷੑ࣭Λ༻͍ͯάϥϑΛѻ͏෼໺ ߦྻࣜ΍ݻ༗஋͕άϥϑͷੑ࣭Λද͢ɽͦͷதͰ΋ڝϓϩʹ࢖͑ͦ͏ͳ΋ͷΛ ঺հ͢Δɽ Rho17 ڝϓϩ෼Պձ Regular ୈ 6 ճ December 12, 2025 17 / 25
18. ڝϓϩ෼Պձ Regular ୈ 6 ճ Rho17 ͸͡Ίʹ ҰൠϚονϯά Tutte-Berge ͷఆཧ εϖΫτϧάϥϑ ཧ࿦ Tutte ߦྻ άϥϑ G(V, E) ʹ͓͍ͯɼล (i, j) ͕ଘࡏ͢ΔͳΒ͹ Aij = xij, Aji = −xij, ͦ ͏Ͱͳ͍ͳΒ͹ Aij = Aji = 0, ·ͨ Aii = 0 ͱͨ͠άϥϑΛ Tutte ߦྻͱ͍͏ɽ Tutte ߦྻͷߦྻࣜ͸׬શϚονϯά͕ଘࡏ͢Δͱ͖ʹݶΓʢଟ߲ࣜͱͯ͠ʣ non-zero Ͱ͋Δɽ ߦྻࣜͷఆٛΛߟ͑Δͱɼ࢒Δ߲͸ด࿏ͷू߹ʹͳΔɽح਺௕ด࿏͕ଘࡏ͢Δ ৔߹ɼͦΕͱٯූ߸ͷ߲΋ݱΕΔͷͰߦྻ͔ࣜΒফ͑ΔɽΑͬͯ࢒Δ߲͸ۮ਺ ௕ͷด࿏ͷΈɼ׬શϚονϯάͷଘࡏ͸໌Β͔ɽ Rho17 ڝϓϩ෼Պձ Regular ୈ 6 ճ December 12, 2025 18 / 25
19. ڝϓϩ෼Պձ Regular ୈ 6 ճ Rho17 ͸͡Ίʹ ҰൠϚονϯά Tutte-Berge ͷఆཧ εϖΫτϧάϥϑ ཧ࿦ Tutte ߦྻ ଟ߲ࣜͷߦྻࣜ͸೉͍͕͠ɼ࣮ࡍ͸ xij ʹཚ਺Λ୅ೖ͠ߦྻࣜΛ਺஋తʹܭࢉ ͢Ε͹ O(N3) Ͱ׬શϚονϯάͷଘࡏΛ൑ఆͰ͖Δɽ ॏΈ෇͖׬શϚονϯάͷ৔߹Ͱ΋΄΅ಉ༷ɽxij Λ xijywij ͱมߋ͠ɼߦྻࣜ ʹ͓͚Δ non-zero ෦෼ͷ y ͷ࠷খ࣍਺͕࠷খॏΈͷ 2 ഒɽ Rho17 ڝϓϩ෼Պձ Regular ୈ 6 ճ December 12, 2025 19 / 25
20. ڝϓϩ෼Պձ Regular ୈ 6 ճ Rho17 ͸͡Ίʹ ҰൠϚονϯά Tutte-Berge ͷఆཧ εϖΫτϧάϥϑ ཧ࿦ ߦྻ໦ఆཧ άϥϑ G(V, E) ͷϥϓϥγΞϯ L ͸ҎԼͷΑ͏ʹఆٛ͞ΕΔɽ Lij =      ௖఺ i ͷ࣍਺ (i = j) −1 × (i ͱ j Λ݁Ϳลͷຊ਺) (i ̸= j) ௖఺ू߹͕ V Ͱล͕ e ͚ͩͷάϥϑͷϥϓϥγΞϯΛ Le ͱ͢Δͱɼ ∑ e∈E Le = L ͕ຬͨ͞ΕΔɽ ͕ͨͬͯ͠ɼx⊤Lx = ∑ {u,v}∈E (xu − xv)2 ΋੒ཱ͢Δɽ Rho17 ڝϓϩ෼Պձ Regular ୈ 6 ճ December 12, 2025 20 / 25
21. ڝϓϩ෼Պձ Regular ୈ 6 ճ Rho17 ͸͡Ίʹ ҰൠϚονϯά Tutte-Berge ͷఆཧ εϖΫτϧάϥϑ ཧ࿦ ߦྻ໦ఆཧ • L ͷ࠷খݻ༗஋͸ 0 • ྻͷ૯࿨͕ 0 ͳͷͰ 1 ͕ݻ༗ϕΫτϧɼݻ༗஋͸ 0 • άϥϑͷୈ 2 ݻ༗஋͕ 0 Ͱ͋Δ͜ͱͱάϥϑ͕ඇ࿈݁Ͱ͋Δ͜ͱ͸ಉ஋ • ୈ 2 ݻ༗஋͸άϥϑͷΫϥελʔੑ΍ฏ໘ඳըͳͲͱ΋ؔ܎͕͋Δ Rho17 ڝϓϩ෼Պձ Regular ୈ 6 ճ December 12, 2025 21 / 25
22. ڝϓϩ෼Պձ Regular ୈ 6 ճ Rho17 ͸͡Ίʹ ҰൠϚονϯά Tutte-Berge ͷఆཧ εϖΫτϧάϥϑ ཧ࿦ ߦྻ໦ఆཧ ߦྻ໦ఆཧ άϥϑͷϥϓϥγΞϯ L ͷ೚ҙͷ༨Ҽࢠ͸ G ͷશҬ໦ͷݸ਺ͱҰக͢Δɽ ܥɿN ௖఺ͷϥϕϧ෇͖໦ͷݸ਺͸ NN−2 Ͱ͋Δɽ׬શάϥϑʹରͯ͠ߦྻ໦ ఆཧΛ༻͍Δ͜ͱͰಘΒΕΔɽ Rho17 ڝϓϩ෼Պձ Regular ୈ 6 ճ December 12, 2025 22 / 25
23. ڝϓϩ෼Պձ Regular ୈ 6 ճ Rho17 ͸͡Ίʹ ҰൠϚονϯά Tutte-Berge ͷఆཧ εϖΫτϧάϥϑ ཧ࿦ LGV ެࣜ ֤ลʹॏΈͷ෇͍ͨ DAG ͕༩͑ΒΕɼ௖఺ͷ૊ (a1, · · · , an, b1, · · · , bn) ͕ࢦఆ ͞ΕΔɽύεͷॏΈΛͦͷύε্ʹ͋Δลͷ૯ੵͱͯ͠ఆٛ͢Δɽ Gij Λʮai → bj ʹࢸΔܦ࿏શͯʹର͢ΔॏΈͷ૯࿨ ʯͱ͢Δɽద߹৚݅ʮi < k ͔ͭ j > l ͳΒ͹ ai → bj ʹࢸΔܦ࿏ͱ ak → bl ʹࢸΔܦ࿏͸ඞͣަΘΔʯΛຬ ͨ࣌͢ɼai → bi ΁ࢸΔޓ͍ʹަΘΒͳ͍ܦ࿏શମʹର͢ΔॏΈͷ૯࿨͸ G(A, B) = det Gij ͱͯ͠ٻΊΒΕΔɽ͜ΕΛ Lindstr¨ om-Gessel-Viennot(LGV) ެࣜͱ͍͏ɽ ಛʹશͯͷลͷॏΈ͕ 1 ͷ৔߹ɼඇަࠩܦ࿏ͷ௨Γ਺ͱͳΔɽ Rho17 ڝϓϩ෼Պձ Regular ୈ 6 ճ December 12, 2025 23 / 25
24. ڝϓϩ෼Պձ Regular ୈ 6 ճ Rho17 ͸͡Ίʹ ҰൠϚονϯά Tutte-Berge ͷఆཧ εϖΫτϧάϥϑ ཧ࿦ ࿅श໰୊ • ARC190-E (?, ຊ൪ AC3 ਓ) • ABC253-Ex (2758) • ABC216-H (3295) Rho17 ڝϓϩ෼Պձ Regular ୈ 6 ճ December 12, 2025 24 / 25
25. ڝϓϩ෼Պձ Regular ୈ 6 ճ Rho17 ͸͡Ίʹ ҰൠϚονϯά Tutte-Berge ͷఆཧ εϖΫτϧάϥϑ ཧ࿦ ࣍ճ༧ࠂ ͍ͭʹ͠Α͏ & Կ͠Α͏...... Rho17 ڝϓϩ෼Պձ Regular ୈ 6 ճ December 12, 2025 25 / 25