https://youtu.be/CJG6t2iBQFI

1. ήʔϜཧ࿦#4*$ୈճ
ଓɾਔ

2. ఏ‫ܞ‬ͷෆຬ ෮श
഑෼͕ड༰తͰ͋Δ͜ͱͷఆٛ ෮श
ਔͷఆٛ ෮श
ਓήʔϜʹ͓͚Δਔͷྫ

3. ఏ‫ܞ‬ͷෆຬ
ఆٛɿఏ‫ܞ‬

ͷෆຬ
ಛੑؔ਺‫ܗ‬ήʔϜ

ʹ͓͍ͯ
഑෼

ͱఏ‫ܞ‬

ΛͱΔͱ͖

Λ഑෼

ʹର͢Δఏ‫ܞ‬

ͷෆຬͱ͍͏

ʹର֤ͯ͠ఏ‫ܞ‬

ͷෆຬ

Λ
େ͖ͳ΋ͷ͔Βॱʹฒ΂ͨϕΫτϧΛ

ͱද͢
ͨͩ͠
͢΂ͯͷ഑෼ʹରͯ͠

ͱ

ʹؔͯ͠͸ෆຬ͸ͳͷͰআ֎ͯ͠ߟ͑Δ
Ώ͑ʹ

͸

‫ݸ‬ͷෆຬΛฒ΂ͨϕΫτϧʹͳΔ
S
(N, v) x ∈
𝒜
(v) S ⊆ N
e(S, x) = v(S) −
∑
i∈S
xi x S
x ∈
𝒜
(v) S ⊆ N e(S, x)
θ(x)
N ∅
θ(x) 2n
− 2

4. ഑෼͕ड༰తͰ͋Δ͜ͱͷఆٛ
ఆٛɿड༰త
̎ͭͷ഑෼

ʹରͯ͠
͋Δ

͕ଘࡏͯ͠



ͱͳΔͱ͖

͸

ΑΓ΋ड༰తͰ͋Δͱ͍͍

ͱද͢

ͷҙຯɿ


ͩͱ

ͱͳΔͨΊ


Ұఆ
x, y ∈
𝒜
(v) k, 1 ≤ k ≤ 2n
− 4
θl(x) = θl(y) ∀l = 1,⋯, k, θk+1(x) θk+1(y)
x y x ≫ y
2n
− 4 θl(x) = θl(y) ∀l = 1,⋯,2n
− 3 θ2n−2(x) = θ2n−2(y)
n−2
∑
l=1
θl(x) =
∑
S⊂N,S≠N,∅
(
v(S) −
∑
i∈S
xi)
=
∑
S⊂N,S≠N,∅
v(S) −
∑
S⊂N,S≠N,∅
∑
i∈S
xi
=
∑
S⊂N,∅
v(S) −
n−1
∑
|S|=1
n−1C|S|−1 ∑
i∈N
xi =
∑
S⊂N,∅
v(S) −
n−1
∑
|S|=1
n−1C|S|−1v(N)

5. ਔͷఆٛ
ఆٛɿਔ
ͦΕΑΓ΋ड༰తͳ഑෼ͷଘࡏ͠ͳ͍഑෼ͷશମΛਔͱ͍͏
ఏ‫ؒܞ‬ͷෆຬΛͰ͖Δ͚ͩ‫ۉ‬౳Խ͢Δͱ͍͏ߟ͑ʹ‫ͮ͘ج‬ղ
ਔͷϙΠϯτ
ఆཧɿΧʔωϧʹඞͣ‫·ؚ‬ΕΔ
ਔ

Χʔωϧ

ަবू߹
ఆཧɿਔ͸ඞͣଘࡏ͢Δ
ఆཧɿਔ͸Ұ఺ͷΈ‫ؚ‬Ήू߹ͱͳΔ
⊆ ⊆

6. ਓήʔϜʹ͓͚Δਔͷྫ

ίΞ͸

v({1,2,3}) = v({1,2}) = 10, v({1,3}) = 7, v({2,3}) = 0, v({1}) = v({2}) = v({3}) = 0
C(v) = {x ∈
𝒜
(v)|x1 + x2 = 10, x2 ≤ 3, x3 = 0}
1
2 3
ίΞ
x1 = 10
x2 = 3
x3 = 0
ަবू߹
Χʔωϧ

7. ਓήʔϜʹ͓͚Δਔͷྫ

ίΞ͸

v({1,2,3}) = v({1,2}) = 10, v({1,3}) = 7, v({2,3}) = 0, v({1}) = v({2}) = v({3}) = 0
C(v) = {x ∈
𝒜
(v)|x1 + x2 = 10, x2 ≤ 3, x3 = 0}

ΛͱΔͱ
֤ఏ‫ܞ‬ͷෆຬ͸






x = (x1, x2, x3) ∈
𝒜
(v)
e({1,2}, x) = v({1,2}) − (x1 + x2) = 10 − (x1 + x2)
e({1,3}, x) = v({1,3}) − (x1 + x3) = 7 − (x1 + x3)
e({2,3}, x) = v({2,3}) − (x2 + x3) = 0 − (x2 + x3)
e({1}, x) = v({1}) − x1 = 0 − x1
e({2}, x) = v({2}) − x2 = 0 − x2
e({3}, x) = v({3}) − x3 = 0 − x3

8. ਓήʔϜʹ͓͚Δਔͷྫ

ίΞ͸

v({1,2,3}) = v({1,2}) = 10, v({1,3}) = 7, v({2,3}) = 0, v({1}) = v({2}) = v({3}) = 0
C(v) = {x ∈
𝒜
(v)|x1 + x2 = 10, x2 ≤ 3, x3 = 0}
഑෼ͷશମ߹ཧੑ͔Β

Λߟྀ͢Δͱ






x1 + x2 + x3 = 10
e({1,2}, x) = 10 − (x1 + x2) = x3
e({1,3}, x) = 7 − (x1 + x3) = x2 − 3
e({2,3}, x) = 0 − (x2 + x3) = x1 − 10
e({1}, x) = − x1
e({2}, x) = − x2
e({3}, x) = − x3

9. ਓήʔϜʹ͓͚Δਔͷྫ

ίΞ͸

v({1,2,3}) = v({1,2}) = 10, v({1,3}) = 7, v({2,3}) = 0, v({1}) = v({2}) = v({3}) = 0
C(v) = {x ∈
𝒜
(v)|x1 + x2 = 10, x2 ≤ 3, x3 = 0}
ਔ͸·ͣ࠷େෆຬΛ࠷খʹ͢Δͷ͔ͩΒ
֤ෆຬΛ

Ͱ͓͑͞

Λ࠷খʹ͢Δ

TU








M M
min M
e({1,2}, x) = x3 ≤ M
e({1,3}, x) = x2 − 3 ≤ M
e({2,3}, x) = x1 − 10 ≤ M
e({1}, x) = − x1 ≤ M
e({2}, x) = − x2 ≤ M
e({3}, x) = − x3 ≤ M
x1 + x2 + x3 = 10 x1, x2, x3 ≥ 0

TU





min M
−M ≤ x1 ≤ M + 10
−M ≤ x2 ≤ M + 3
−M ≤ x3 ≤ M
x1 + x2 + x3 = 10 x1, x2, x3 ≥ 0

10. 
TU






͕ଘࡏ͢ΔͨΊͷ

ͷ৚݅͸

͕ଘࡏ͢ΔͨΊʹ͸


͕ଘࡏ͢ΔͨΊʹ͸


͕ଘࡏ͢ΔͨΊʹ͸


৚݅Λຬͨ͢࠷খͷ

͸
Ώ͑ʹ

min M
−M ≤ x1 ≤ M + 10 −M ≤ x2 ≤ M + 3 −M ≤ x3 ≤ M
x1 + x2 + x3 = 10 x1, x2, x3 ≥ 0
x1, x2, x3 M
x1 M ≥ − 5
x2 M ≥ −
3
2
x3 M ≥ 0
−3M ≤ x1 + x2 + x3 = 10 ≤ 13 + 3M ⇔ −
10
3
≤ M, − 1 ≤ M
M x3 = 0
ਓήʔϜʹ͓͚Δਔͷྫ

ίΞ͸

v({1,2,3}) = v({1,2}) = 10, v({1,3}) = 7, v({2,3}) = 0, v({1}) = v({2}) = v({3}) = 0
C(v) = {x ∈
𝒜
(v)|x1 + x2 = 10, x2 ≤ 3, x3 = 0}
0
M
−5
−
10
3
−1
−
3
2

11. ਓήʔϜʹ͓͚Δਔͷྫ

ίΞ͸

v({1,2,3}) = v({1,2}) = 10, v({1,3}) = 7, v({2,3}) = 0, v({1}) = v({2}) = v({3}) = 0
C(v) = {x ∈
𝒜
(v)|x1 + x2 = 10, x2 ≤ 3, x3 = 0}
࠷େෆຬΛ࠷খʹ͢Δͷ͸









M = 0
e({1,2}, x) = x3 = 0
e({1,3}, x) = x2 − 3 ≤ 0
e({2,3}, x) = x1 − 10 ≤ 0
e({1}, x) = − x1 ≤ 0
e({2}, x) = − x2 ≤ 0
e({3}, x) = − x3 = 0
x1 + x2 + 0 = 10 x1, x2 ≥ 0




0 ≤ x1 ≤ 10
0 ≤ x2 ≤ 3
x3 = 0
x1 + x2 = 10

12. ਓήʔϜʹ͓͚Δਔͷྫ

ίΞ͸

v({1,2,3}) = v({1,2}) = 10, v({1,3}) = 7, v({2,3}) = 0, v({1}) = v({2}) = v({3}) = 0
C(v) = {x ∈
𝒜
(v)|x1 + x2 = 10, x2 ≤ 3, x3 = 0}
ਔ͸ೋ൪໨ʹେ͖͍ෆຬΛ࠷খʹ͢Δͷ͔ͩΒ
֤ෆຬΛ

Ͱ͓͑͞

Λ࠷খʹ͢Δ

TU








M′

M = 0 M′

min M′

e({1,2}, x) = x3 ≤ M
e({1,3}, x) = x2 − 3 ≤ M′

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

e({1}, x) = − x1 ≤ M′

e({2}, x) = − x2 ≤ M′

e({3}, x) = − x3 ≤ M
x1 + x2 + 0 = 10 x1, x2 ≥ 0, x3 = 0

TU




min M′

−M′

≤ x1 ≤ M′

+ 10
−M′

≤ x2 ≤ M′

+ 3 ⇔ − M′

≤ 10 − x1 ≤ M′

+ 3
0 ≤ x1 ≤ 10
M′

0

13. 
TU




੍໿৚݅Λ੔ཧ͢Δͱ
ҎԼͷؔ܎ʹͳΔ

Αͬͯ

͕ଘࡏ͢ΔͨΊʹ͸

Ώ͑ʹ৚݅Λຬͨ͢࠷খͷ

͸

͕ͨͬͯ͠


min M′

−M′

≤ x1 ≤ M′

+ 10
−M′

≤ 10 − x1 ≤ M′

+ 3 ⇔ 7 − M′

≤ x1 ≤ M′

+ 10
0 ≤ x1 ≤ 10
M′

0
0 − M′

7 − M′

≤ x1 ≤ 10 + M′

10
x1 M′

≥ −
3
2
M′

−
3
2
x1 =
17
2
𝒩
(v) = {x ∈
𝒜
(v)|(x1, x2, x3) = (
17
2
,
3
2
,0)}
ਓήʔϜʹ͓͚Δਔͷྫ

ίΞ͸

v({1,2,3}) = v({1,2}) = 10, v({1,3}) = 7, v({2,3}) = 0, v({1}) = v({2}) = v({3}) = 0
C(v) = {x ∈
𝒜
(v)|x1 + x2 = 10, x2 ≤ 3, x3 = 0}

14. ਓήʔϜʹ͓͚Δਔͷྫ

ίΞ͸

v({1,2,3}) = v({1,2}) = 10, v({1,3}) = 7, v({2,3}) = 0, v({1}) = v({2}) = v({3}) = 0
C(v) = {x ∈
𝒜
(v)|x1 + x2 = 10, x2 ≤ 3, x3 = 0}
1
2 3
ίΞ
x1 = 10
x2 = 3
x3 = 0
ަবू߹
Χʔωϧ
ਔ

15. ਓήʔϜʹ͓͚Δਔͷྫ

ίΞ͸

v({1,2,3}) = v({1,2}) = v({1,3}) = v({2,3}) = 10, v({1}) = v({2}) = v({3}) = 0
C(v) = ∅
1
2 3
x1 = 0
x2 = 0
x3 = 0

(x1, x2, x3) =
(
10
3
,
10
3
,
10
3 )
ަবू߹͔ͭΧʔωϧ

16. ਓήʔϜʹ͓͚Δਔͷྫ

ίΞ͸

v({1,2,3}) = v({1,2}) = v({1,3}) = v({2,3}) = 10, v({1}) = v({2}) = v({3}) = 0
C(v) = ∅

ΛͱΔͱ
֤ఏ‫ܞ‬ͷෆຬ͸






x = (x1, x2, x3) ∈
𝒜
(v)
e({1,2}, x) = v({1,2}) − (x1 + x2) = 10 − (x1 + x2)
e({1,3}, x) = v({1,3}) − (x1 + x3) = 10 − (x1 + x3)
e({2,3}, x) = v({2,3}) − (x2 + x3) = 10 − (x2 + x3)
e({1}, x) = v({1}) − x1 = 0 − x1
e({2}, x) = v({2}) − x2 = 0 − x2
e({3}, x) = v({3}) − x3 = 0 − x3

17. ਓήʔϜʹ͓͚Δਔͷྫ

ίΞ͸

v({1,2,3}) = v({1,2}) = v({1,3}) = v({2,3}) = 10, v({1}) = v({2}) = v({3}) = 0
C(v) = ∅
഑෼ͷશମ߹ཧੑ͔Β

Λߟྀ͢Δͱ






x1 + x2 + x3 = 10
e({1,2}, x) = 10 − (x1 + x2) = x3
e({1,3}, x) = 10 − (x1 + x3) = x2
e({2,3}, x) = 10 − (x2 + x3) = x1
e({1}, x) = − x1
e({2}, x) = − x2
e({3}, x) = − x3

18. ਓήʔϜʹ͓͚Δਔͷྫ

ίΞ͸

v({1,2,3}) = v({1,2}) = v({1,3}) = v({2,3}) = 10, v({1}) = v({2}) = v({3}) = 0
C(v) = ∅
ਔ͸·ͣ࠷େෆຬΛ࠷খʹ͢Δͷ͔ͩΒ
֤ෆຬΛ

Ͱ͓͑͞

Λ࠷খʹ͢Δ

TU








M M
min M
e({1,2}, x) = x3 ≤ M
e({1,3}, x) = x2 ≤ M
e({2,3}, x) = x1 ≤ M
e({1}, x) = − x1 ≤ M
e({2}, x) = − x2 ≤ M
e({3}, x) = − x3 ≤ M
x1 + x2 + x3 = 10 x1, x2, x3 ≥ 0

19. ਓήʔϜʹ͓͚Δਔͷྫ

ίΞ͸

v({1,2,3}) = v({1,2}) = v({1,3}) = v({2,3}) = 10, v({1}) = v({2}) = v({3}) = 0
C(v) = ∅

TU






͕ଘࡏ͢ΔͨΊͷ

ͷ৚݅͸


৚݅Λຬͨ͢࠷খͷ

͸

Ώ͑ʹ



શମ߹ཧੑΛߟྀ͢Ε͹

min M
−M ≤ x1 ≤ M −M ≤ x2 ≤ M −M ≤ x3 ≤ M
x1 + x2 + x3 = 10 x1, x2, x3 ≥ 0
x1, x2, x3 M M ≥ 0
−3M ≤ x1 + x2 + x3 = 10 ≤ 3M ⇔ −
10
3
≤ M,
10
3
≤ M
M
10
3
−
10
3
≤ x1 ≤
10
3
−
10
3
≤ x2 ≤
10
3
−
10
3
≤ x3 ≤
10
3
x1 = x2 = x3 =
10
3
10
3
M
−
10
3
0
࠷େͰͳ͍ͱશମ߹ཧੑΛຬͨ͞ͳ͍

20. ਓήʔϜʹ͓͚Δਔͷྫ

ίΞ͸

v({1,2,3}) = v({1,2}) = v({1,3}) = v({2,3}) = 10, v({1}) = v({2}) = v({3}) = 0
C(v) = ∅
1
2 3
x1 = 0
x2 = 0
x3 = 0

(x1, x2, x3) =
(
10
3
,
10
3
,
10
3 )
ަবू߹͔ͭΧʔωϧ ͔ͭਔ

21. ਔͷఆٛ
ఆٛɿਔ
ͦΕΑΓ΋ड༰తͳ഑෼ͷଘࡏ͠ͳ͍഑෼ͷશମΛਔͱ͍͏
ఏ‫ؒܞ‬ͷෆຬΛͰ͖Δ͚ͩ‫ۉ‬౳Խ͢Δͱ͍͏ߟ͑ʹ‫ͮ͘ج‬ղ
ਔͷϙΠϯτ
ఆཧɿΧʔωϧʹඞͣ‫·ؚ‬ΕΔ
ਔ

Χʔωϧ

ަবू߹
ఆཧɿਔ͸ඞͣଘࡏ͢Δ
ఆཧɿਔ͸Ұ఺ͷΈ‫ؚ‬Ήू߹ͱͳΔ
⊆ ⊆

22. ήʔϜཧ࿦#4*$ୈճ
ଓɾਔ
࣍ճɿਔʹؔ͢Δఆཧͷূ໌