https://youtu.be/THr_gZ2HbW4

1. ήʔϜཧ࿦#4*$ୈճ
ਔ

2. ఏ‫ܞ‬ͷෆຬ
഑෼͕ड༰తͰ͋Δ͜ͱͷఆٛ
ਔͷఆٛ
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
ྫ

഑෼


ͱ͢Δͱ


Ͱ͋Δ͔Β

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
v({1,2,3}) = v({1,2}) = v({1,3}) = 10, v({2,3}) = 0, v({1}) = v({2}) = v({3}) = 0
x = (4,1,5) y = (1,5,4)
θ(x) = (5,1, − 1, − 4, − 5, − 6)
θ(y) = (5,4, − 1, − 4, − 5, − 9)
x ≫ y






v({1,2}) − (1 + 5) = 10 − 6 = 4
v({1,3}) − (1 + 4) = 10 − 5 = 5
v({2,3}) − (5 + 4) = 0 − 9 = − 9
v({1}) − 1 = − 1
v({2}) − 5 = − 5
v({3}) − 4 = − 4






v({1,2}) − (4 + 1) = 10 − 5 = 5
v({1,3}) − (4 + 5) = 10 − 9 = 1
v({2,3}) − (1 + 5) = 0 − 6 = − 6
v({1}) − 4 = − 4
v({2}) − 1 = − 1
v({3}) − 5 = − 5

5. ഑෼͕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)

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

Χʔωϧ

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

7. ਔͷఆٛ
ఆٛɿਔ
ͦΕΑΓ΋ड༰తͳ഑෼ͷଘࡏ͠ͳ͍഑෼ͷશମΛਔͱ͍͏
ਔʹଐ͢Δ഑෼͸
ฒ΂ͨෆຬͷதͰ΋ͬͱ΋େ͖ͳෆຬ͕
࠷΋খ͘͞ͳΔ΋ͷͰͳ͚Ε͹ͳΒͳ͍
্‫ه‬ෆຬ͕ෳ਺͋Δͱ͖͸
ฒ΂ͨෆຬͷதͰ൪໨ʹେ͖ͳෆຬ͕
࠷΋খ͘͞ͳΔ΋ͷͰͳ͚Ε͹ͳΒͳ͍
্‫ه‬ෆຬ͕ෳ਺͋Δͱ͖͸
ฒ΂ͨෆຬͷதͰ൪໨ʹେ͖ͳෆຬ͕
࠷΋খ͘͞ͳΔ΋ͷͰͳ͚Ε͹ͳΒͳ͍
ɾɾɾ



θ(x) = (6,1, − 1, − 4, − 5, − 6)
θ(y) = (5,1, − 1, − 4, − 5, − 6)
θ(z) = (5,4, − 1, − 4, − 5, − 9)



θ(x) = (6,1, − 1, − 4, − 5, − 6)
θ(y) = (5,1, − 1, − 4, − 5, − 6)
θ(z) = (5,4, − 1, − 4, − 5, − 9)



θ(x) = (6,1, − 1, − 4, − 5, − 6)
θ(y) = (5,1, − 1, − 4, − 5, − 6)
θ(z) = (5,4, − 1, − 4, − 5, − 9)

8. ਓήʔϜʹ͓͚Δਔͷྫ
1
2 3
ίΞ

{(10, 0, 0)}
ަবू߹

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)}
Χʔωϧ

9. ਓήʔϜʹ͓͚Δਔͷྫ

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)}

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) = 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

10. ਓήʔϜʹ͓͚Δਔͷྫ

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

11. ਓήʔϜʹ͓͚Δਔͷྫ

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)}
ਔ͸·ͣ࠷େෆຬΛ࠷খʹ͢Δͷ͔ͩΒ
֤ෆຬΛ

Ͱ͓͑͞

Λ࠷খʹ͢Δ

TU








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

TU





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

12. 
TU






͕ଘࡏ͢ΔͨΊͷ

ͷ৚݅͸

͕ଘࡏ͢ΔͨΊʹ͸


͕ଘࡏ͢ΔͨΊʹ͸


৚݅Λຬͨ͢࠷খͷ

͸
Ώ͑ʹ

͜ͷͨΊ

ɿਔ͸

min M
−M ≤ x1 ≤ M + 10 −M ≤ x2 ≤ M −M ≤ x3 ≤ M
x1 + x2 + x3 = 10 x1, x2, x3 ≥ 0
x1, x2, x3 M
x1 M ≥ − 5
x2, x3 M ≥ 0
−3M ≤ x1 + x2 + x3 = 10 ≤ 10 + 3M ⇔ −
10
3
≤ M, 0 ≤ M
M x2 = x3 = 0 x1 = 10
𝒩
(v) = {(10,0,0)}
ਓήʔϜʹ͓͚Δਔͷྫ

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)}
0
M
−5
−
10
3

13. ਓήʔϜʹ͓͚Δਔͷྫ
1
2 3
ίΞ

{(10, 0, 0)}
ަবू߹

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)}
Χʔωϧ
ਔ

14. ήʔϜཧ࿦#4*$ୈճ
ਔ
࣍ճɿଓɾਔ