一対多マッチング

1. ήʔϜཧ࿦#4*$ୈճ
ҰରଟϚονϯά

2. ҰରଟϚονϯά
Ϟσϧ
‫ݸ‬ਓ߹ཧੑ
҆ఆੑ
۩ମྫ

3. ҰରҰϚονϯάͱҰରଟϚονϯά͕͋Δ
ҰରҰϚονϯά͸ਓΛਓʹׂΓ౰ͯ
ྫ ݁ࠗ
ҰରଟϚονϯά͸ ෳ਺ਓΛͭͷϞϊ ‫ۀا‬ ‫ࣨڀݚ‬ පӃ FUD ʹׂΓ౰ͯ
ྫ ब৬ ‫ࣨڀݚ‬બͼ ‫ݚ‬मҩ ֶߍબ୒
ҰରଟϚονϯά
ෳ਺ਓ‫͑ޏ‬Δ

4. Ϟσϧ
ֶੜΛ‫ׂʹࣨڀݚ‬Γ౰ͯΔϞσϧΛߟ͑Δ
ֶੜͷू߹Λ ‫ࣨڀݚ‬ͷू߹Λ ͱ͢Δ
ͱ͢Δ
ศٓతʹ ֶੜ͸ఴࣈ Λ෇༩͠ ‫ࣨڀݚ‬͸ఴࣈ Λ෇༩͢Δ
ֶੜ ͸ ্ʹબ޷ Λ΋ͪ
‫ࣨڀݚ‬ ͸ ্ʹબ޷ Λ΋ͭ
۩ମྫ
ֶੜʹ͍ͭͯ ͱॻ͍ͨ৔߹
͕΋ͬͱ΋๬·͍͠ ॴଐ͍ͨ͠ ‫͓ͯ͑ߟͱࣨڀݚ‬Γ ࣍ʹ๬·͍͠‫ࣨڀݚ‬͸
΍ͷ‫ॴʹࣨڀݚ‬ଐ͢Δ͘Β͍ͳΒແॴଐ ‫࣮ݱ‬తͰ͸ͳ͍͕ Ͱ͍͍ͨ ͱղऍ͢Δ
‫͍ͯͭʹࣨڀݚ‬ ͱॻ͍ͨ৔߹
͸ॴଐ͍͕ͤͨ͞ ͸ॴଐͤͨ͘͞ͳ͍ ͱղऍ͢Δ
M W
m = |M| 0 w = |W| 0
1,2,⋯, m m + 1,m + 2,⋯, m + w
i ∈ M W∪{i} ≿i (i = 1,2,⋯, m)
j ∈ W M∪{j} ≿j (j = m + 1,m + 2,⋯, m + w)
M = {1,2,3} W = {4,5,6,7}
≿1: 4 7 1 6 5
≿4: 1 2 4 3

5. Ϟσϧ
͢΂ͯͷֶੜ͓Αͼ‫ࣨڀݚ‬ͷબ޷Λ Λฒ΂ͨ΋ͷΛ
ͱ͢Δ
·ͨ ֤‫ࣨڀݚ‬ ͸ΩϟύγςΟ Λ΋ͪ ͸੔਺Ͱ Λຬͨ͢ͱ͢Δ
ͭ·Γ গͳ͘ͱ΋ਓ͸ॴଐͤ͞Δ͜ͱ͕Ͱ͖ ਓ·Ͱॴଐͤ͞Δ͜ͱ͕Ͱ͖Δ
͜ͷબ޷૊͕༩͑ΒΕͨ‫ֶͰݩ‬ੜͷॴଐઌΛಋؔ͘਺Λ Ͱද͢
͜ͷ ͕Ϛονϯάͱͯ͠ຬͨ͢΂͖৚݅͸
ɾֶ֤ੜ ʹ͍ͭͯ ͳ͓ ͸ॴଐ͍ͯ͠ͳ͍ঢ়ଶͰ͋Δ
ɾ‫߸ه‬ ʹΑΓ ʹॴଐ͢Δֶੜͷਓ਺Λද͢΋ͷͱ͢Δ͜Ε͕ Λຬͨ͢
͸୭΋ॴଐ͍ͯ͠ͳ͍ঢ়ଶͰ͋Δ
≿i (i = 1,2,⋯, m, m + 1,m + 2,⋯, m + w)
≿ = ( ≿1 , ≿2 , ⋯, ≿m , ≿m+1 , ≿m+2 , ⋯, ≿m+w )
j ∈ W cj cj cj ≥ 1
cj
μ : M → M ∪ W
μ
i ∈ M μ(i) = j ∈ W ∪ {i} μ(i) = i
|μ(j)| j ∈ W 0 ≤ |μ(j)| ≤ cj
|μ(j)| = 0
ֶ֤ੜΛͲ͜ʹׂΓ౰ͯΔ͔ͱ͍͏ؔ਺

6. ‫ݸ‬ਓ߹ཧੑ
‫ݸ‬ਓ߹ཧੑ
Ϛονϯά ͕ ͷ΋ͱͰ‫ݸ‬ਓ߹ཧతͰ͋Δͱ͸
Ͱ͋Δ͢΂ͯͷ ʹ͍ͭͯ ͔ͭ ͕੒Γཱͭ͜ͱͰ͋Δ
͸ֶੜ ͸‫ࣨڀݚ‬ ʹॴଐ͢Δ͜ͱʹͳΔ
͜ͷͱ͖ Ͱ͋Δ͜ͱ͸ ʹͱͬͯແॴଐͰ͋ΔΑΓ ʹॴଐ͢Δ͜ͱ͕๬·্͍͠
Ͱ͋Δ͜ͱ͸ ʹͱͬͯ Λ‫ڋ‬൱͢ΔΑΓॴଐͤͨ͞ํ͕๬·͍͠ͱ͍͏͜ͱͰ͋Δ
μ ≿ = ( ≿1 , ≿2 , ⋯, ≿m , ≿m+1 , ≿m+2 , ⋯, ≿m+w )
μ(i) = j i ∈ M, j ∈ W j ≻i i i ≻j j
μ(i) = j i j
j ≻i i i j
i ≻j j j i

7. ҆ఆੑ
҆ఆੑ
Ϛονϯά ͕ ͷ΋ͱͰ҆ఆͰ͋Δͱ͸
͕‫ݸ‬ਓ߹ཧతͯ͋Δ
Ͱ͋ΔΑ͏ͳ ʹ͍ͭͯߟ͑Δ
΋͠ ͳΒ͹ ͳΔ͢΂ͯͷ ʹ͍ͭͯ Ͱ͋Δ
΋͠ ͳΒ͹ Ͱ͋Δ
҆ఆੑΛຬͨ͢Ϛονϯά͸ ඞͣଘࡏ͠
ҰରҰͰͷ%ΞϧΰϦζϜͱಉ༷ͷΞϧΰϦζϜͰ‫ٻ‬ΊΔ͜ͱ͕Ͱ͖Δ
μ ≿
μ
j ≻i μ(i) i ∈ M, j ∈ W
|μ(j)| = cj μ(k) = j k ∈ M k ≻j i
|μ(j)| cj j ≻j i ΑΓ๬·ֶ͍͠ੜ͕ॴଐ
ΩϟύγςΟ͸௒͑ͯͳ͍͕
͜ͷֶੜ͸ॴଐͤͨ͘͞ͳ͍
͸ׂΓ౰ͯΒΕͨ‫ࣨڀݚ‬ ΑΓ
๬·͍͠‫ࣨڀݚ‬ Λ‫ر‬๬
i μ(i)
j

8. ۩ମྫ
ֶੜ ≿1: 5 7 6 1
ֶੜ ≿2: 5 6 7 2
ֶੜ ≿3: 7 6 5 3
‫ࣨڀݚ‬ ≿5: 4 3 2 1 5
‫ࣨڀݚ‬ ≿6: 2 3 4 1 5
‫ࣨڀݚ‬ ≿7: 1 4 2 3 7
ֶੜ ≿4: 5 6 4 7
c5 = 2
c6 = 2
c7 = 1

9. ۩ମྫ
ֶੜ ≿2: 5 6 7 2
ֶੜ ≿3: 7 6 5 3
‫ࣨڀݚ‬ ≿5: 4 3 2 1 5
‫ࣨڀݚ‬ ≿6: 2 3 4 1 6
‫ࣨڀݚ‬ ≿7: 1 4 2 3 7
ֶੜ ≿4: 5 6 4 7
c5 = 2
c6 = 2
c7 = 1
๬·͍͠‫ࣨڀݚ‬Λࢦఆ
ֶੜ ≿1: 5 7 6 1

10. ۩ମྫ
ֶੜ ≿2: 5 6 7 2
ֶੜ ≿3: 7 6 5 3
‫ࣨڀݚ‬ ≿5: 4 3 2 1 5
‫ࣨڀݚ‬ ≿6: 2 3 4 1 6
‫ࣨڀݚ‬ ≿7: 1 4 2 3 7
ֶੜ ≿4: 5 6 4 7
c5 = 2
c6 = 2
c7 = 1
ΩϟύγςΟ·Ͱ͸Ωʔϓ
ֶੜ͸அΒΕΔ
ֶੜ͸Ωʔϓ
ֶੜ ≿1: 5 7 6 1

11. ۩ମྫ
ֶੜ ≿2: 5 6 7 2
ֶੜ ≿3: 7 6 5 3
‫ࣨڀݚ‬ ≿5: 4 3 2 1 5
‫ࣨڀݚ‬ ≿6: 2 3 4 1 6
‫ࣨڀݚ‬ ≿7: 1 4 2 3 7
ֶੜ ≿4: 5 6 4 7
c5 = 2
c6 = 2
c7 = 1
ֶੜ ≿1: 5 7 6 1
ࢦఆՄೳͳதͰ
࠷΋๬·͍͠‫ࣨڀݚ‬Λࢦఆ

12. ۩ମྫ
ֶੜ ≿2: 5 6 7 2
ֶੜ ≿3: 7 6 5 3
‫ࣨڀݚ‬ ≿5: 4 3 2 1 5
‫ࣨڀݚ‬ ≿6: 2 3 4 1 6
‫ࣨڀݚ‬ ≿7: 1 4 2 3 7
ֶੜ ≿4: 5 6 4 7
c5 = 2
c6 = 2
c7 = 1
ֶੜ ≿1: 5 7 6 1
ΩϟύγςΟͳͷͰਓͷΈΩʔϓ
બ޷ॱংͷߴֶ͍ੜΛΩʔϓ
ֶੜ͸அΒΕΔ

13. ۩ମྫ
ֶੜ ≿2: 5 6 7 2
ֶੜ ≿3: 7 6 5 3
‫ࣨڀݚ‬ ≿5: 4 3 2 1 5
‫ࣨڀݚ‬ ≿6: 2 3 4 1 6
‫ࣨڀݚ‬ ≿7: 1 4 2 3 7
ֶੜ ≿4: 5 6 4 7
c5 = 2
c6 = 2
c7 = 1
ֶੜ ≿1: 5 7 6 1
ࢦఆՄೳͳதͰ
࠷΋๬·͍͠‫ࣨڀݚ‬Λࢦఆ
ΩϟύγςΟతʹ΋
બ޷తʹ΋அΔཧ༝͸ͳ͍

14. ۩ମྫ
ֶੜ ≿2: 5 6 7 2
ֶੜ ≿3: 7 6 5 3
‫ࣨڀݚ‬ ≿5: 4 3 2 1 5
‫ࣨڀݚ‬ ≿6: 2 3 4 1 6
‫ࣨڀݚ‬ ≿7: 1 4 2 3 7
ֶੜ ≿4: 5 6 4 7
c5 = 2
c6 = 2
c7 = 1
ֶੜ ≿1: 5 7 6 1
͜ͷखॱͰ‫ٻ‬ΊͨϚονϯά͸҆ఆͰ͋Δֶ֤ੜ ‫ݸ͕ࣨڀݚ‬ਓ߹ཧੑΛຬͨ͢ͷ͸༰қʹ֬ೝͰ͖Δ
·ͨ ֶੜʹ͍ͭͯ Ͱ͋Δ͕ Ͱ͋Γ ͳͷͰ
‫ͯؔ͠ʹࣨڀݚ‬ ͱͳ͍ͬͯΔֶੜʹ͍ͭͯ΋৚݅Λຬͨ͢
5 ≻1 μ(1) = 7 |μ(5)| = c5 = 2 μ(2) = 5 μ(4) = 5
2 ≻5 1 4 ≻5 1
҆ఆੑ
Ϛονϯά ͕ ͷ΋ͱͰ҆ఆͰ͋Δͱ͸
͕‫ݸ‬ਓ߹ཧతͯ͋Δ
Ͱ͋ΔΑ͏ͳ ʹ͍ͭͯߟ͑Δ
΋͠ ͳΒ͹ ͳΔ͢΂ͯͷ ʹ͍ͭͯ Ͱ͋Δ΋͠ ͳΒ͹ Ͱ͋Δ
μ ≿
μ
j ≻i μ(i) i ∈ M, j ∈ W
|μ(j)| = cj μ(k) = j k ∈ M k ≻j i |μ(j)| cj j ≻j i

15. ήʔϜཧ࿦#4*$ୈճ
ҰରଟϚονϯά