ෆڠ࿨Իͷͳ͍ύʔτ΁ͷׂ౰ –࠷దͳׂ౰͔Β҆ఆͳׂ౰΁– ҏ౾Ӭ ༸Ұ 2020/01/30 (last update: 2020/01/31) Optimization Night #2

ࣗݾ঺հ ҏ౾Ӭ ༸Ұʢ͍ͣͳ͕ Α͏͍ͪʣ ॴଐ ਆಸ઒େֶ ޻ֶ෦ʢॿڭʣ ઐ໳ ਺ཧ࠷దԽɼΦϖϨʔγϣϯζɾϦαʔν ݚڀ಺༰ ωοτϫʔΫ্ͷΫϥελϦϯά΍ϥϯΩϯάʹؔ͢Δ ϞσϦϯάͱΞϧΰϦζϜઃܭ 2 / 15

എܠ ࠓճͷൃද͸ɼਆಸ઒େֶ ࡾ޷෣͞Μͷଔۀ࿦จʹؔ͢Δ΋ͷͰ͢ ͳ͓ɼࢲ͸ਧ૗ָ΍Իָʹؔͯ͠͸શ͘ͷૉਓͰ͢... ਧ૗ָ෦ͷݱঢ় • େ͖͘ 12 ύʔτʢ໦؅ 5, ۚ؅ 5ɼݭָث 1ɼଧָث 1ʣʹ෼͚ΒΕɺ ৽ೖ෦һ͸ೖ෦࣌ʹॴଐύʔτΛܾఆ • ίϯΫʔϧͰ͸՝୊ۂʹ͓͚Δָثฤ੒ͷࢦఆ΍ ग़৔ਓ਺ 55 ਓͱ͍͏੍໿͕͋Δ ໰୊఺ • ৽ೖ෦һͷҙࢥʹ͙ͦΘͳ͍ύʔτʹׂΓ౰ͯΔͱɼ Ϟνϕʔγϣϯͷ௿Լ΍࠷ѱͷ৔߹ୀ෦͢ΔՄೳੑ͕͋Δ • ύʔτͷฤ੒ʹภΓ͕͋Δͱԋ૗৔ෆརʹͳͬͯ͠·͏ 3 / 15

എܠ • ֶੜʢ৽ೖ෦һʣ ... ͜Ε·Ͱͷܦݧ౳͔Βر๬ָثʢύʔτʣΛද໌ • ίϯϚεɾύʔτϦʔμʔ ... ֶੜͷܦྺ΍ΦʔσΟγϣϯͳͲΛجʹֶੜʹର͢Δر๬ද໌ • ࢦಋऀ ... ͦΕͧΕͷر๬Λ΋ͱʹύʔτ΁ͷׂ౰ΛٻΊΔ ʢฤ੒ͷόϥϯεɼੈ୅ަ୅ʢ4 ೥ޙͷ෦಺ߏ੒ʣ΋ҙࣝʣ → ҆ఆϚονϯάͷ֓೦Λ༻͍ͯɼΑΓྑ͍ύʔτ΁ͷׂ౰ΛٻΊΔ 4 / 15

࠷దͳׂ౰ vs. ҆ఆͳׂ౰ • N: ֶੜͷू߹ • P: ύʔτͷू߹ • Lj : ύʔτ j ͷఆһͷԼݶ • N(j): ύʔτ j Λر๬͢Δֶੜͷू߹ • P(i): ֶੜ i Λड͚ೖΕՄೳͳύʔτू߹ • Uj : ύʔτ j ͷఆһͷ্ݶ ܾఆม਺ xij = { 1 (ֶੜ i Λύʔτ j ΁ׂΓ౰ͯΔ) 0 (ͦΕҎ֎). ʢ࠷దԽΞϓϩʔνʣ ޮ༻ʢຬ଍౓ʣuij ͕༩͑ΒΕ্ͨͰɼ ԿΒ͔ͷ໨తؔ਺Λ࠷దԽ max. f(uij , xij ) s.t. ∑ j∈P (i) xij ≤ 1 (i ∈ N), Lj ≤ ∑ i∈N(j) xij ≤ Uj (j ∈ P). ʢ҆ఆϚονϯάʹΑΔΞϓϩʔνʣ ֶੜͱύʔτ͸ɼͦΕͧΕʹର͢Δر๬ Λॱংʢ⪰ʣͰද໌ s.t. ∑ j∈P (i) xij ≤ 1 (i ∈ N), Lj ≤ ∑ i∈N(j) xij ≤ Uj (j ∈ P), +҆ఆੑΛ՝੍͢໿. 5 / 15

҆ఆϚονϯά • ҆ఆϚονϯά ⇔ ϒϩοΩϯάϖΞ͕͍ͳ͍Ϛονϯά • ϒϩοΩϯάϖΞ ≈ ۦ͚མͪͦ͠͏ͳϖΞ ʢఆٛʣ உੑ i ͱঁੑ j ͕Ϛονϯά x ʹ͓͚Δ ϒϩοΩϯάϖΞ ⇔ (1) i ͱ j ͸ݱࡏϚον͍ͯ͠ͳ͍ (2) i ͸ಠ਎ or ݱࡏͷ૬खΑΓ j Λ޷Ή (3) j ͸ಠ਎ or ݱࡏͷ૬खΑΓ i Λ޷Ή 1 2 3 A B C 1 2 2 1 3 2 1 B A C A B C B A ଟରҰϚονϯάʢஉੑΛֶੜɼঁੑΛύʔτʹஔ͖׵͑ʣ ʹ͓͚ΔϒϩοΩϯάϖΞ... (1), (2) (3’) ύʔτ j ʹ഑ଐ͞Ε͍ͯΔֶੜͷਓ਺͸ఆһͷ্ݶະຬ or ݱࡏ഑ଐ͞Ε͍ͯΔֶੜΑΓ΋ֶੜ i Λ޷Ή 6 / 15

҆ఆϚονϯά • ҆ఆϚονϯά ⇔ ϒϩοΩϯάϖΞ͕͍ͳ͍Ϛονϯά • ϒϩοΩϯάϖΞ ≈ ۦ͚མͪͦ͠͏ͳϖΞ ʢఆٛʣ உੑ i ͱঁੑ j ͕Ϛονϯά x ʹ͓͚Δ ϒϩοΩϯάϖΞ ⇔ (1) i ͱ j ͸ݱࡏϚον͍ͯ͠ͳ͍ (2) i ͸ಠ਎ or ݱࡏͷ૬खΑΓ j Λ޷Ή (3) j ͸ಠ਎ or ݱࡏͷ૬खΑΓ i Λ޷Ή 1 2 3 A B C 1 2 2 1 3 2 1 B A C A B C B A ଟରҰϚονϯάʢஉੑΛֶੜɼঁੑΛύʔτʹஔ͖׵͑ʣ ʹ͓͚ΔϒϩοΩϯάϖΞ... (1), (2) (3’) ύʔτ j ʹ഑ଐ͞Ε͍ͯΔֶੜͷਓ਺͸ఆһͷ্ݶະຬ or ݱࡏ഑ଐ͞Ε͍ͯΔֶੜΑΓ΋ֶੜ i Λ޷Ή 6 / 15

҆ఆϚονϯά • ҆ఆϚονϯά ⇔ ϒϩοΩϯάϖΞ͕͍ͳ͍Ϛονϯά • ϒϩοΩϯάϖΞ ≈ ۦ͚མͪͦ͠͏ͳϖΞ ʢఆٛʣ உੑ i ͱঁੑ j ͕Ϛονϯά x ʹ͓͚Δ ϒϩοΩϯάϖΞ ⇔ (1) i ͱ j ͸ݱࡏϚον͍ͯ͠ͳ͍ (2) i ͸ಠ਎ or ݱࡏͷ૬खΑΓ j Λ޷Ή (3) j ͸ಠ਎ or ݱࡏͷ૬खΑΓ i Λ޷Ή 1 2 3 A B C 1 2 2 1 3 2 1 B A C A B C B A ଟରҰϚονϯάʢஉੑΛֶੜɼঁੑΛύʔτʹஔ͖׵͑ʣ ʹ͓͚ΔϒϩοΩϯάϖΞ... (1), (2) (3’) ύʔτ j ʹ഑ଐ͞Ε͍ͯΔֶੜͷਓ਺͸ఆһͷ্ݶະຬ or ݱࡏ഑ଐ͞Ε͍ͯΔֶੜΑΓ΋ֶੜ i Λ޷Ή 6 / 15

҆ఆϚονϯά • ҆ఆϚονϯά ⇔ ϒϩοΩϯάϖΞ͕͍ͳ͍Ϛονϯά • ϒϩοΩϯάϖΞ ≈ ۦ͚མͪͦ͠͏ͳϖΞ ʢఆٛʣ உੑ i ͱঁੑ j ͕Ϛονϯά x ʹ͓͚Δ ϒϩοΩϯάϖΞ ⇔ (1) i ͱ j ͸ݱࡏϚον͍ͯ͠ͳ͍ (2) i ͸ಠ਎ or ݱࡏͷ૬खΑΓ j Λ޷Ή (3) j ͸ಠ਎ or ݱࡏͷ૬खΑΓ i Λ޷Ή 1 2 3 A B C 1 2 2 1 3 2 1 B A C A B C B A ۦ͚མͪ ଟରҰϚονϯάʢஉੑΛֶੜɼঁੑΛύʔτʹஔ͖׵͑ʣ ʹ͓͚ΔϒϩοΩϯάϖΞ... (1), (2) (3’) ύʔτ j ʹ഑ଐ͞Ε͍ͯΔֶੜͷਓ਺͸ఆһͷ্ݶະຬ or ݱࡏ഑ଐ͞Ε͍ͯΔֶੜΑΓ΋ֶੜ i Λ޷Ή 6 / 15

҆ఆϚονϯά • ҆ఆϚονϯά ⇔ ϒϩοΩϯάϖΞ͕͍ͳ͍Ϛονϯά • ϒϩοΩϯάϖΞ ≈ ۦ͚མͪͦ͠͏ͳϖΞ ʢఆٛʣ உੑ i ͱঁੑ j ͕Ϛονϯά x ʹ͓͚Δ ϒϩοΩϯάϖΞ ⇔ (1) i ͱ j ͸ݱࡏϚον͍ͯ͠ͳ͍ (2) i ͸ಠ਎ or ݱࡏͷ૬खΑΓ j Λ޷Ή (3) j ͸ಠ਎ or ݱࡏͷ૬खΑΓ i Λ޷Ή 1 2 3 A B C 1 2 2 1 3 2 1 B A C A B C B A ଟରҰϚονϯάʢஉੑΛֶੜɼঁੑΛύʔτʹஔ͖׵͑ʣ ʹ͓͚ΔϒϩοΩϯάϖΞ... (1), (2) (3’) ύʔτ j ʹ഑ଐ͞Ε͍ͯΔֶੜͷਓ਺͸ఆһͷ্ݶະຬ or ݱࡏ഑ଐ͞Ε͍ͯΔֶੜΑΓ΋ֶੜ i Λ޷Ή 6 / 15

੔਺ઢܗ࠷దԽ໰୊ʢILPʣͱͯ͠ͷఆࣜԽ max. ∑ i∈N ∑ j∈P (i) xij ϖΞ਺࠷େԽ s.t. ∑ j∈P (i) xij ≤ 1 (i ∈ N), Lj ≤ ∑ i∈N(j) xij ≤ Uj (j ∈ P), Uj ≤ Uj ∑ q∈P ⪰ j (i) xiq + ∑ p∈N⪰ i (j) xpj ϒϩοΩϯάϖΞΛഉআ (i ∈ N, j ∈ P(i)). • P⪰ j (i) ֶੜ i ͕ύʔτ j ͱಉఔ౓Ҏ্ʹر๬͢Δύʔτͷू߹ • N⪰ i (j) ύʔτ j ֶ͕ੜ i ͱಉఔ౓Ҏ্ʹड͚ೖΕͯྑ͍ͱߟ͑Δֶੜͷू߹ ⇡ P⌫( )= { } ⇡ N⌫( )= { } 7 / 15

ҰൠʹఆһͷԼݶ੍໿Λ՝͢ͱ҆ఆϚονϯά͕ଘࡏ͢Δͱ͸ݶΒͳ͍ 1 → ଘࡏ͠ͳ͍৔߹ʹ΋ɼԿΒ͔ͷҙຯͰ҆ఆϚονϯάʹׂ͍ۙ౰ΛٻΊ͍ͨ → ҆ఆϚονϯάʹׂ͍ۙ౰ ≈ ϒϩοΩϯάϖΞ਺͕࠷খͷϚονϯά ܾఆม਺ yij = { 1 (ֶੜ i ͱύʔτ j ͕ϒϩοΩϯάϖΞͱͳΔ) 0 (ͦΕҎ֎). max. a ∑ i∈N ∑ j∈P (i) xij ϖΞ਺࠷େԽ −b ∑ i∈N ∑ j∈P (i) yij ϒϩοΩϯάϖΞ਺࠷খԽ s.t. ∑ j∈P (i) xij ≤ 1 (i ∈ N), Lj ≤ ∑ i∈N(j) xij ≤ Uj (j ∈ P), (1 − yij )Uj ≤ Uj ∑ q∈P ⪰ j (i) xiq + ∑ p∈N⪰ i (j) xpj (i ∈ N, j ∈ P(i)). 1্ݶ੍໿͚ͩͰ͋Ε͹ɼৗʹ҆ఆϚονϯά͸ଘࡏ͢Δ 8 / 15

ܭࢉػ࣮ݧʢ1ʣ σʔλʢر๬Ϧετ, ఆһͷ্Լݶʣ • ر๬Ϧετ... ਧ૗ָ෦ʹॴଐ͢Δֶ෦ 1 ೥ੜʢ36 ਓʣ & ίϯϚε΁ͷΞϯέʔτʹج͖ͮ࡞੒ • ఆһͷԼݶ... ύʔτ಺ͷηΫγϣϯʹ 1 ਓͣͭ഑ஔͨ͠৔߹ͷਓ਺ • ఆһͷ্ݶ... ίϯΫʔϧʹग़৔Մೳͳ࠷େਓ਺ ܭࢉػ؀ڥ • PC: Core i5 @3.4GHz, 40GB memory • MIP solver: Python API of Gurobi 8.0 Fl Ob Fg Cl Sax Tp Hr Tb Euph Tuba Cb Perc set of parts 0 2 4 6 8 10 12 # of students minimum quota maximum quota (a,b)=(0.5,0.5) • ܭࢉ࣌ؒ: 0.016 ඵ • ҆ఆϚονϯά͕ಘΒΕͨ • શͯͷֶੜ͸ɼୈ 2 ر๬Ҏ্ͷ ύʔτʹׂΓ౰ͯΒΕͨ • ΫϥϦωοτύʔτ͸ఆһ্ݶΑ Γᴴ͔ʹগͳ͍ਓ਺ׂ͕Γ౰ͯΒ Ε͍ͯΔ → ਧ૗ָͷಛੑ্ɼશମͷ໿ 40% Λ઎ΊΔ͜ͱ͕ཧ૝త → ฤ੒ͷόϥϯε͕ෆे෼ 9 / 15

ฤ੒ͷόϥϯε΋ߟྀͨ͠ఆࣜԽ ഑ଐਓ਺Λఆһͷ্ԼݶʹऩΊΔ͚ͩͰͳ͘ฤ੒ͷόϥϯε΋ߟྀ͍ͨ͠ → ֤ύʔτʹ഑ଐ਺ͷ໨ඪ஋ Tj Λઃఆ 2 → εϥοΫม਺ S+ j , S− j Λಋೖ͠ɼ഑ଐ਺͕ Tj ʹͰ͖ΔݶΓۙ͘ͳΔΑ͏ఆࣜԽ max. a ∑ i∈N ∑ j∈P (i) xij ϖΞ਺࠷େԽ −b ∑ i∈N ∑ j∈P (i) yij ϒϩοΩϯάϖΞ਺࠷খԽ −c ∑ j∈P (S+2 j + S−2 j ) ໨ඪ஋ͱͷζϨΛ࠷খԽ s.t. ∑ j∈P (i) xij ≤ 1 (i ∈ N), Lj ≤ ∑ i∈N(j) xij ≤ Uj (j ∈ P), (1 − yij )Uj ≤ Uj ∑ q∈P ⪰ j (i) xiq + ∑ p∈N⪰ i (j) xpj (i ∈ N, j ∈ P(i)), (Tj − ∑ i∈N(j) xij ) + S+ j − S− j = 0 (j ∈ P). 2ࠓճ͸্Լݶͷதԝ஋ʹઃఆ 10 / 15

ܭࢉػ࣮ݧʢ2ʣ Fl Ob Fg Cl Sax Tp Hr Tb Euph Tuba Cb Perc set of parts 0 2 4 6 8 10 12 # of students minimum quota maximum quota (a,b)=(0.5,0.5) → Fl Ob Fg Cl Sax Tp Hr Tb Euph Tuba Cb Perc set of parts 0 2 4 6 8 10 12 # of students minimum quota maximum quota (a,b,c)=(0.1,0.5,0.4) • ܭࢉ࣌ؒ: 0.05 ඵ • ϒϩοΩϯάϖΞ਺͸ 6 • શͯͷֶੜ͸ɼୈ 3 ر๬Ҏ্ͷύʔτʹׂΓ౰ͯΒΕͨ • ઌͷ݁Ռͱൺ΂Δͱɼฤ੒ͷόϥϯε΋վળ 11 / 15

·ͱΊ • ҆ఆϚονϯάΛٻΊΔ໰୊ͷ ILP ఆࣜԽΛ঺հ → ৽ͨʹ௥Ճ͢΂੍͖໿ʢཁ݅ʣ͕ੜͨ͡ࡍ΋ରԠ͕༰қ → ൚༻࠷దԽιϧόʔͰٻղՄೳ • ਧ૗ָ෦ʹ͓͚Δύʔτ΁ͷׂ౰ʹద༻ • ֶੜͱύʔτ૒ํͷر๬ͱฤ੒ͷόϥϯεΛߟׂྀͨ͠౰ΛٻΊͨ 12 / 15

ࢀߟจݙ • ٱอװ༤ɼాଜ໌ٱɼদҪ஌ݾʢฤʣ ɼ ʮԠ༻਺ཧܭըϋϯυϒοΫʯ ɼே૔ॻ ళɼ2002. • M. Baïou and M. Balinski, “The stable admissions polytope,” Mathematical Programming, Ser.A, Vol.87, pp.427–439, 2000. • M. Delorme et al., “Mathematical models for stable matching problems with ties and incomplete lists,” European Journal of Operational Research, Vol.277, pp.426–441, 2019. 13 / 15

ʢଟରҰʣϚονϯάʹ͓͚Δ҆ఆੑ੍໿ ر๬Ϧετʹؔ͢ΔԾఆ • ϖΞʹͳΓͨ͘ͳ͍૬ख͸Ϧετʹॻ͔ͳͯ͘΋ྑ͍ʢෆ׬શϦετʣ • ಉ͘͡Β͍޷Ή૬खʹ͸ಉॱҐΛڐ͢ • ͓ޓ͍ʹر๬Ϧετʹهࡌ͍͋ͬͯ͠ΔϖΞ͚ͩΛر๬Ϧετʹ࢒͢ ʢఆٛʣ ֶੜ i ͱύʔτ j ͕Ϛονϯά x ʹ͓͚ΔϒϩοΩϯάϖΞ ⇔ (1) ֶੜ i ͱύʔτ j ͸ݱࡏϚον͍ͯ͠ͳ͍ (2) ֶੜ i ͸Ͳͷύʔτʹ΋഑ଐ͞Ε͍ͯͳ͍ or ݱࡏͷ഑ଐύʔτΑΓ j Λ޷Ή (3) ύʔτ j ʹ഑ଐ͞Ε͍ͯΔֶੜͷਓ਺͸ఆһͷ্ݶະຬ or ݱࡏ഑ଐ͞Ε͍ͯΔֶੜΑΓ΋ֶੜ i Λ޷Ή ֶੜ i ͱύʔτ j ͕ϒϩοΩϯάϖΞͰͳ͍ ⇔ Uj ≤ Uj ∑ q∈P ⪰ j (i) xiq + ∑ p∈N⪰ i (j) xpj 14 / 15

ʢଟରҰʣϚονϯάʹ͓͚Δ҆ఆੑ੍໿ (ରۮΛࣔ͢) Uj > Uj ∑ q∈P ⪰ j (i) xiq + ∑ p∈N ⪰ i (j) xpj ΛԾఆ͢ΔͱɼҎԼ͕੒Γཱͭɽ ∑ q∈P ⪰ j (i) xiq = 0 (1) ∧ ∑ p∈N ⪰ i (j) xpj < Uj (2). ֶੜ i ͸ߴʑ 1 ͭͷύʔτʹׂ౰ΒΕ͍ͯΔ͜ͱͱࣜ (1) ΑΓɼͲͷύʔτʹ΋ׂ౰ΒΕ͍ͯͳ͍ or P \ P ⪰ j (i) ʹଐ͢ Δύʔτʢj ΑΓ΋޷·͘͠ͳ͍ύʔτʣʹׂ౰ΒΕ͍ͯΔɽ ύʔτ j ʹؔͯ͠͸ɼ഑ଐ͞Ε͍ͯΔֶੜͷਓ਺͕ Uj ະຬͷ৔߹͸ (i, j) ͕ϒϩοΩϯάϖΞͰ͋Δ͜ͱ͸໌Β͔ͳͷ Ͱɼ഑ଐ͞Ε͍ͯΔֶੜ਺͕ Uj ʹҰக͢Δ৔߹ɼͭ·Γ ∑ p∈N(j)\N ⪰ i (j) xpj + ∑ p∈N ⪰ i (j) xpj = Uj (3) ͕੒Γཱͭ৔߹Λߟ͑Δɽࣜ (2) ͱ (3) ͔Βɼύʔτ j ʹগͳ͘ͱ΋ 1 ਓ͸ N(j) \ N⪰ i (j) ʹଐ͢Δֶੜʢi ΑΓ޷· ͘͠ͳֶ͍ੜʣ͕഑ଐ͞Ε͍ͯΔɽ͕ͨͬͯ͠ɼֶੜ i ͱύʔτ j ͸Ϛονϯά x ʹ͓͚ΔϒϩοΩϯάϖΞͰ͋Δɽ ٯʹ͍ͭͯ΋ɼ্هͷٞ࿦ΛԼ͔Β୧Δ͜ͱͰಉ༷ʹࣔͤΔɽ 15 / 15