https://youtu.be/EwTl7r2tqNU

1. ήʔϜཧ࿦#4*$ୈճ
҆ఆू߹

2. ഑෼ͷࢧ഑ ෮श
ίΞͷఆٛ ෮श
҆ఆू߹
ίΞͱ҆ఆू߹
ਓήʔϜʹ͓͚Δ҆ఆू߹ͷྫ

3. ഑෼ͷࢧ഑
ఆٛɿ഑෼ͷࢧ഑
̎ͭͷ഑෼

ͱఏ‫ܞ‬

ΛͱΔ࣍ͷ̎৚͕݅ຬͨ͞ΕΔͱ͖

͸

Λ

Λ௨ͯ͠ࢧ഑͢Δͱ͍͍

ͱॻ͘


̎ͭͷ഑෼

ʹରͯ͠
͋Δఏ‫ܞ‬

͕ଘࡏͯ͠

͕੒Γཱͭͱ͖
୯ʹ

͸

Λࢧ഑͢Δͱ͍͍

ͱॻ͘
x, y S ⊆ N
x y S x domS y
v(S) ≥
∑
i∈S
xi
xi yi, ∀i ∈ S
x, y S x domS y
x y x dom y

4. ίΞͷఆٛ
ఆٛɿίΞ
ಛੑؔ਺‫ܗ‬ήʔϜ

ͷίΞͱ͸
ଞͷ͍͔ͳΔ഑෼ʹ΋ࢧ഑͞Εͳ͍഑෼ͷू߹Ͱ͋Δ

ͳΔ

͕ଘࡏ͠ͳ͍

ίΞʹ‫·ؚ‬ΕΔ഑෼ͦΕͧΕ͕
্‫ه‬ͷࢧ഑ؔ܎·ͨ͸ఏ‫߹ܞ‬ཧੑΛຬͨ͢
(N, v)
C(v) = {x ∈
𝒜
(v)|y dom x y }

5. ҆ఆू߹
ఆٛ
ಛੑؔ਺‫ܗ‬ήʔϜ

ʹ͓͍ͯ
഑෼ͷू߹

͕҆ఆू߹Ͱ͋Δͱ͸
ҎԼΛຬͨ͢ͱ͖Λ͍͏
֎෦҆ఆੑ

ʹଐ͞ͳ͍͢΂ͯͷ഑෼

ʹؔͯ͠
͋Δ

಺ͷ഑෼

͕ଘࡏͯ͠

͕

Λࢧ഑͢Δ
಺෦҆ఆੑ

಺ͷ೚ҙͷ഑෼

ʹ͍ͭͯ‫ʹ͍ޓ‬ଞΛࢧ഑͠ͳ͍
ิ
ίΞ͸഑෼ʹରͯ͠ఆ͕ٛͨ͠
҆ఆू߹͸഑෼ͷ෦෼ू߹ʹରͯ͠ఆٛ͢Δ
ิ
ϑΥϯϊΠϚϯϞϧήϯγϡςϧϯղͱ΋‫ݺ‬͹ΕΔ
(N, v) K
K y
K x x y
K x, y

6. ίΞͱ҆ఆू߹
ఆཧ
ಛੑؔ਺‫ܗ‬ήʔϜ

ʹ͓͍ͯ
ίΞͱ҆ఆू߹͕‫ʹڞ‬ଘࡏ͢ΔͳΒ͹
ίΞ͸҆ఆू߹ʹ‫·ؚ‬ΕΔ

Λࣔͤ͹Α͍
೚ҙͷ

ΛͱΔ͜ͷͱ͖

Ͱ͋Δ͜ͱΛࣔͤ͹Α͍

ͱ͢Δͱ

͸͋Δ഑෼

ʹࢧ഑͞ΕΔ͜ͱʹͳΔͨΊ

͕ίΞͷ഑෼Ͱ͋Δ͜ͱʹໃ६
(N, v)
C(v) ⊆ K
x ∈ C(v) x ∈ K
x ∉ K x x′

∈ K
x ֎෦҆ఆੑ

ʹଐ͞ͳ͍͢΂ͯͷ഑෼

ʹؔͯ͠
͋Δ

಺ͷ഑෼

͕ଘࡏͯ͠

͕

Λࢧ഑͢Δ
K y
K x x y

7. ਓήʔϜʹ͓͚Δ҆ఆू߹ͷྫ

ίΞ͸

ϓϨΠϠʔҎ֎΋ߩ‫ͯ͠ݙ‬ΔͷʹಘΒΕΔ഑෼͸
ϓϨΠϠʔ
͸ಉҙͰ͖Δͷ͔ʁ
v({1,2,3}) = v({1,2}) = v({1,3}) = 10, v({2,3}) = 0, v({1}) = v({2}) = v({3}) = 0
C(v) = {x ∈
𝒜
(v)|(x1, x2, x3) = (10,0,0)}
1
2 3
x1 ≤ 10
x2 ≤ 0
x3 ≤ 0

8. ਓήʔϜʹ͓͚Δ҆ఆू߹ͷྫ
഑෼

ͷࢧ഑͢ΔྖҬΛਤࣔ
x
1
2 3
x


v(S) ≥
∑
i∈S
xi
xi yi, ∀i ∈ S
഑෼

͕ఏ‫ܞ‬

Λ௨ͯ͠ࢧ഑͠͏ΔྖҬᶃ

x {2,3}
xi yi, ∀i ∈ {2,3}
ᶃ

9. ਓήʔϜʹ͓͚Δ҆ఆू߹ͷྫ
഑෼

ͷࢧ഑͢ΔྖҬΛਤࣔ
x
1
2 3
x


v(S) ≥
∑
i∈S
xi
xi yi, ∀i ∈ S
഑෼

͕ఏ‫ܞ‬

Λ௨ͯ͠ࢧ഑͠͏ΔྖҬᶃ

x {2,3}
xi yi, ∀i ∈ {2,3}
ᶃ ഑෼

͕ఏ‫ܞ‬

Λ௨࣮ͯ͠ࡍʹࢧ഑͢Δʹ͸

Λຬͨ͢௚ઢ͕

ͷԼํʹ͋Δඞཁ͕͋Δ ࠨਤࢀর
x {2,3}
x1 = v({1,2,3}) − v({2,3})
x

10. ਓήʔϜʹ͓͚Δ҆ఆू߹ͷྫ
഑෼

ͷࢧ഑͢ΔྖҬΛਤࣔ
x
1
2 3
x


v(S) ≥
∑
i∈S
xi
xi yi, ∀i ∈ S
഑෼

͕ఏ‫ܞ‬

Λ௨ͯ͠ࢧ഑͠͏ΔྖҬᶃ

x {2,3}
xi yi, ∀i ∈ {2,3}
ᶃ
ᶄ
഑෼

͕ఏ‫ܞ‬

Λ௨ͯ͠
ࢧ഑͠͏ΔྖҬᶄ
x {1,3}

11. ਓήʔϜʹ͓͚Δ҆ఆू߹ͷྫ
഑෼

ͷࢧ഑͢ΔྖҬΛਤࣔ
x
1
2 3
x


v(S) ≥
∑
i∈S
xi
xi yi, ∀i ∈ S
഑෼

͕ఏ‫ܞ‬

Λ௨ͯ͠ࢧ഑͠͏ΔྖҬᶃ

x {2,3}
xi yi, ∀i ∈ {2,3}
ᶃ
ᶄ
഑෼

͕ఏ‫ܞ‬

Λ௨ͯ͠
ࢧ഑͠͏ΔྖҬᶄ
x {1,3}
഑෼

͕ఏ‫ܞ‬

Λ௨ͯ͠
ࢧ഑͠͏ΔྖҬᶅ
x {1,2}
ᶅ

12. ਓήʔϜʹ͓͚Δ҆ఆू߹ͷྫ

ίΞ͸

v({1,2,3}) = v({1,2}) = v({1,3}) = 10, v({2,3}) = 0, v({1}) = v({2}) = v({3}) = 0
C(v) = {x ∈
𝒜
(v)|(x1, x2, x3) = (10,0,0)}
1
2 3
ίΞʹ‫·ؚ‬ΕΔ఺͸҆ఆू߹ʹ‫·ؚ‬ΕΔ͕
͜ͷ఺͚ͩͰ͸ଞͷ഑෼Λࢧ഑Ͱ͖ͳ͍


v(S) ≥
∑
i∈S
xi
xi yi, ∀i ∈ S

13. ਓήʔϜʹ͓͚Δ҆ఆू߹ͷྫ

ίΞ͸

v({1,2,3}) = v({1,2}) = v({1,3}) = 10, v({2,3}) = 0, v({1}) = v({2}) = v({3}) = 0
C(v) = {x ∈
𝒜
(v)|(x1, x2, x3) = (10,0,0)}
1
2 3
೚ҙͷઢ෼ΛҾ͘ 1
2 3
͜ͷઢ෼ʹ‫·ؚ‬ΕΔ఺ʹ
ࢧ഑͞ΕΔྖҬ


v(S) ≥
∑
i∈S
xi
xi yi, ∀i ∈ S
͜ͷઢ෼্ͷ఺ʹؔͯ͠
ࢧ഑ؔ܎͸ͳ͍

͕ͩ

x2 x′

2, x3 x′

3
v({2,3}) = 0 x2 + x3
x1 ≤ 10
(x1, x2, x3)
(x′

1, x′

2, x′

3)
಺෦҆ఆੑ
֎෦҆ఆੑ
x2 ≤ 10

14. ਓήʔϜʹ͓͚Δ҆ఆू߹ͷྫ

ίΞ͸

v({1,2,3}) = v({1,2}) = v({1,3}) = 10, v({2,3}) = 0, v({1}) = v({2}) = v({3}) = 0
C(v) = {x ∈
𝒜
(v)|(x1, x2, x3) = (10,0,0)}
1
2 3
೚ҙͷઢ෼ΛҾ͘
1
2 3
͜ͷઢ෼ʹ‫·ؚ‬ΕΔ఺ʹ
ࢧ഑͞ΕΔྖҬ


v(S) ≥
∑
i∈S
xi
xi yi, ∀i ∈ S
͜ͷઢ෼্ͷ఺ʹؔͯ͠
ࢧ഑ؔ܎͸ͳ͍
x1 ≤ 10
಺෦҆ఆੑ
֎෦҆ఆੑ

15. ਓήʔϜʹ͓͚Δ҆ఆू߹ͷྫ

ίΞ͸

v({1,2,3}) = v({1,2}) = v({1,3}) = 10, v({2,3}) = 0, v({1}) = v({2}) = v({3}) = 0
C(v) = {x ∈
𝒜
(v)|(x1, x2, x3) = (10,0,0)}
1
2 3


v(S) ≥
∑
i∈S
xi
xi yi, ∀i ∈ S
಺෦҆ఆੑ
֎෦҆ఆੑ
҆ఆू߹
҆ఆू߹͸࿈ଓͰ͋Δඞཁ͕͋Δ
੾Ε໨͕͋Δͱࢧ഑͠ͳ͍ྖҬ͕ଘࡏ
͢ͳΘͪ֎෦҆ఆੑΛຬͨ͞ͳ͍
1
2 3

16. ਓήʔϜʹ͓͚Δ҆ఆू߹ͷྫ

ίΞ͸

v({1,2,3}) = v({1,2}) = v({1,3}) = 10, v({2,3}) = 0, v({1}) = v({2}) = v({3}) = 0
C(v) = {x ∈
𝒜
(v)|(x1, x2, x3) = (10,0,0)}
1
2 3
҆ఆू߹
‫ۂ‬ઢ෦෼Λަব‫ۂ‬ઢͱ͍͏
ϓϨΠϠʔ
ͷ‫ͨ·ྗڠ‬͸݁ୗʹΑΓ
ίΞͷ഑෼Ҏ֎ʹ৽ͨʹղͱͯ͠ަব‫ۂ‬ઢ෦෼͕ՃΘΔ
͜Ε͸ϓϨΠϠʔͷಠ઎Λ‫
ͣ͞ڐ‬
ϓϨΠϠʔ
ͷަবྗ΍݁ୗͷ‫ʹ͞ڧ‬ΑΓ
རಘ഑෼͕ఆ·Δ͜ͱΛࣔࠦ͢Δ

17. ήʔϜཧ࿦#4*$ୈճ
҆ఆू߹
࣍ճɿଓɾ҆ఆू߹