https://youtu.be/9wXESD9rEsk

1. ήʔϜཧ࿦#4*$ୈճ
ଓɾަবू߹

2. ҟٞ ෮श
‫ٯ‬ҟٞ ෮श
ަবू߹ͷఆٛͱίΞͱͷؔ܎ ෮श
ਓήʔϜʹ͓͚Δަবू߹ͷྫ

3. ҟٞ
ަবΛҟٞɾ‫ٯ‬ҟٞͱ͍͏‫Ͱܗ‬ఆࣜԽ
ఆٛɿҟٞ
ಛੑؔ਺‫ܗ‬ήʔϜ

ʹ͓͍ͯ
഑෼

ͱਓͷϓϨΠϠʔ

ΛͱΔ
ࠓ


ͳΔఏ‫ܞ‬

ͱ഑෼

͕ଘࡏͯ͠


ͷ̎ͭͷ৚͕݅ຬͨ͞ΕͯΔͱ͢Δ
͜ͷͱ͖഑෼

ʹ͓͍ͯϓϨΠϠʔ

͸

ʹରͯ͠ҟٞ

Λ΋ͭͱ͍͏
(N, v)
x ∈
𝒜
(v) i, j ∈ N
i ∈ S j ∉ S S ⊆ N y ∈
𝒜
(v)
∑
k∈S
yk ≤ v(S)
yk xk, k ∈ S
x i j (y, S)

4. ‫ٯ‬ҟٞ
ަবΛҟٞɾ‫ٯ‬ҟٞͱ͍͏‫Ͱܗ‬ఆࣜԽ
ఆٛɿ‫ٯ‬ҟٞ

ͷ

ʹର͢Δҟٞ

ʹରͯ͠


ͳΔఏ‫ܞ‬

ͱ഑෼

͕ଘࡏͯ͠



ͷͭͷ৚͕݅ຬͨ͞ΕͯΔͱ͢Δ
͜ͷͱ͖ϓϨΠϠʔ

͸

ͷҟٞ

ʹରͯ͠‫ٯ‬ҟٞ

Λ΋ͭͱ͍͏
i j (y, S)
i ∉ T j ∈ T T ⊆ N z ∈
𝒜
(v)
∑
k∈T
zk ≤ v(T)
zk ≥ xk, k ∈ TS
zk ≥ yk, k ∈ T ∩ S
j i (y, S) (z, T)

5. ަবू߹ͷఆٛ
ఆٛɿަবू߹
഑෼

ʹ͓͍ͯ
ͲͷϓϨΠϠʔ΋ҟٞΛ΋ͨͳ͍͔
ͳ͍͠͸ҟٞΛ΋ͭϓϨΠϠʔ͕͍ͨͱͯ͠΋
ͦΕʹର͢Δ‫ٯ‬ҟ͕ٞଘࡏ͢Δͱ͖
഑෼

͸҆ఆͰ͋Δͱ͍͍
҆ఆͳ഑෼ͷશମΛަবू߹ͱ͍͏
ήʔϜ

ʹ͓͚Δަবू߹Λ

Ͱද͢
x ∈
𝒜
(v)
x
(N, v) ℬ(v)
ఆཧɿίΞͱަবू߹ͷؔ܎
ήʔϜ

ʹ͓͍ͯ
ίΞΛ

ަবू߹Λ

ͱ͢Δͱ

(N, v) C(v) ℬ(v) C(v) ⊆ ℬ(v)

6. ਓήʔϜʹ͓͚Δަবू߹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
ίΞ

{(10, 0, 0)}

7. ਓήʔϜʹ͓͚Δަবू߹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
ίΞ

{(10, 0, 0)}
ަবू߹

(x1 + ϵ, ϵ′

, x3 + ϵ′

′

)


ΑΓ


͸
ϓϨΠϠʔͷϓϨΠϠʔʹର͢ΔҟٞʹͳΔ
x1 + ϵ + x3 + ϵ′

′

≤ v({1,3}) = 10
x1 x1 + ϵ, x3 x3 + ϵ′

′

(x1 + ϵ, ϵ′

, x3 + ϵ′

′

) {1,3}
ҟٞ


∑
k∈S
yk ≤ v(S)
yk xk, k ∈ S
ϓϨΠϠʔͷ‫ٯ‬ҟٞ͸ଘࡏ͢Δ͔ʁ/0



Λ

Λຬͨ͢Α͏ʹͱΔͱ


Ͱ͋Γ


ͱͳΔͨΊ
x2 0, x3 + ϵ′

′

0
v({2}) = v({2,3}) = 0
z = (z1, z2, z3) z2 ≥ x2 0, z3 ≥ x3 + ϵ′

′

0
T = {2}
∑
k∈{2}
zk v({2}) = 0
T = {2,3}
∑
k∈{2,3}
zk v({2,3}) = 0
‫ٯ‬ҟٞ



∑
k∈T
zk ≤ v(T)
zk ≥ xk, k ∈ T S
zk ≥ yk, k ∈ T ∩ S

(x1, x2, x3)

8. ਓήʔϜʹ͓͚Δަবू߹ͷྫɿίΞ͕ଘࡏ

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
ίΞͷ഑෼͕ަবू߹ͱҰக͢Δ
‫ٯ‬ҟٞ



∑
k∈T
zk ≤ v(T)
zk ≥ xk, k ∈ T S
zk ≥ yk, k ∈ T ∩ S
ҟٞ


∑
k∈S
yk ≤ v(S)
yk xk, k ∈ S
ަবू߹

9. ਓήʔϜʹ͓͚Δަবू߹ͷྫɿίΞ͕ଘࡏ

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

ΛίΞʹଐ͞ͳ͍഑෼ͱ͠
͜Ε͕ෆ҆ఆͰ͋Δ͜ͱΛࣔ͢
ίΞʹଐ͞ͳ͍ͨΊ

͸

·ͨ͸

Λຬͨ͢

ͷͱ͖

Ͱ͋Δ

ΛͱΔ
ͨͩ͠

͸

Λຬͨ͢ਖ਼ͷ࣮਺


ΑΓ


͸
ϓϨΠϠʔͷϓϨΠϠʔʹର͢ΔҟٞʹͳΔ
(x′

1, x′

2, x′

3)
(x′

1, x′

2, x′

3) x′

1 + x′

2 10 x′

1 + x′

2 = 10, x′

2 3
x′

1 + x′

2 10 x′

3 0 (y′

1, y′

2, y′

3) = (x′

1 + ϵ, x′

2 + ϵ′

, ϵ′

′

)
ϵ, ϵ′

, ϵ′

′

ϵ + ϵ′

+ ϵ′

′

= 10 − (x′

1 + x′

2) 0
y′

1 + y′

2 = x′

1 + ϵ + x′

2 + ϵ′

v({1,2}) = 10
x′

1 x′

1 + ϵ = y′

1, x′

2 x′

2 + ϵ′

′

= y′

2
(y′

1, y′

2, y′

3) {1,2}

(y′

1, y′

2, y′

3)
‫ٯ‬ҟٞ



∑
k∈T
zk ≤ v(T)
zk ≥ xk, k ∈ T S
zk ≥ yk, k ∈ T ∩ S
ҟٞ


∑
k∈S
yk ≤ v(S)
yk xk, k ∈ S

(x′

1, x′

2, x′

3)

10. ਓήʔϜʹ͓͚Δަবू߹ͷྫɿίΞ͕ଘࡏ

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

ΛίΞʹଐ͞ͳ͍഑෼ͱ͠
͜Ε͕ෆ҆ఆͰ͋Δ͜ͱΛࣔ͢
ίΞʹଐ͞ͳ͍ͨΊ

͸

·ͨ͸

Λຬͨ͢

ͷͱ͖

Ͱ͋Δ

ΛͱΔ
লུ


͸
ϓϨΠϠʔͷϓϨΠϠʔʹର͢ΔҟٞʹͳΔ
͔͠͠

ΛͱΔ
ͨͩ͠

͸


Λຬͨ͢ਖ਼ͷ࣮਺



͜ΕΑΓϓϨΠϠʔ͸͜ͷҟٞʹରͯ͠
‫ٯ‬ҟٞΛ΋ͭ
(x′

1, x′

2, x′

3)
(x′

1, x′

2, x′

3) x′

1 + x′

2 10 x′

1 + x′

2 = 10, x′

2 3
x′

1 + x′

2 10 x′

3 0 (y′

1, y′

2, y′

3) = (x′

1 + ϵ, x′

2 + ϵ′

, ϵ′

′

)
(y′

1, y′

2, y′

3) {1,2}
(z′

1, z′

2, z′

3) = (x′

1 + ϵ̄, ϵ̄′

, x′

3 + ϵ̄′

′

)
ϵ̄, ϵ̄′

, ϵ̄′

′

ϵ̄ + ϵ̄′

+ ϵ̄′

′

= 7 − (x′

1 + x′

3) 0 ϵ̄ ≥ ϵ
z′

1 + z′

3 = x′

1 + ϵ̄ + x′

3 + ϵ̄′

v({1,3}) = 7
x′

3 x′

x + ϵ̄ = z′

3 x′

1 x′

1 + ϵ = y′

1 ≤ x′

1 + ϵ̄ = z′

1

(y′

1, y′

2, y′

3)
‫ٯ‬ҟٞ



∑
k∈T
zk ≤ v(T)
zk ≥ xk, k ∈ T S
zk ≥ yk, k ∈ T ∩ S
ҟٞ


∑
k∈S
yk ≤ v(S)
yk xk, k ∈ S

(x′

1, x′

2, x′

3)

(z′

1, z′

2, z′

3)

11. ਓήʔϜʹ͓͚Δަবू߹ͷྫɿίΞ͕ଘࡏ

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

ΛίΞʹଐ͞ͳ͍഑෼ͱ͠
͜Ε͕ෆ҆ఆͰ͋Δ͜ͱΛࣔ͢
ίΞʹଐ͞ͳ͍ͨΊ

͸

·ͨ͸

Λຬͨ͢

ͷͱ͖

Ͱ͋Δ

ΛͱΔ
ͨͩ͠

͸

Λຬͨ͢ਖ਼ͷ࣮਺


ΑΓ


͸
ϓϨΠϠʔͷϓϨΠϠʔʹର͢ΔҟٞʹͳΔ
(x′

1, x′

2, x′

3)
(x′

1, x′

2, x′

3) x′

1 + x′

2 10 x′

1 + x′

2 = 10, x′

2 3
x′

1 + x′

2 10 x′

3 0 (y′

1, y′

2, y′

3) = (x′

1 + ϵ, x′

2 + ϵ′

, ϵ′

′

)
ϵ, ϵ′

, ϵ′

′

ϵ + ϵ′

+ ϵ′

′

= 10 − (x′

1 + x′

2) 0
y′

1 + y′

2 = x′

1 + ϵ + x′

2 + ϵ′

v({1,2}) = 10
x′

1 x′

1 + ϵ = y′

1, x′

2 x′

2 + ϵ′

′

= y′

2
(y′

1, y′

2, y′

3) {1,2}

(y′

1, y′

2, y′

3)
‫ٯ‬ҟٞ



∑
k∈T
zk ≤ v(T)
zk ≥ xk, k ∈ T S
zk ≥ yk, k ∈ T ∩ S
ҟٞ


∑
k∈S
yk ≤ v(S)
yk xk, k ∈ S

(x′

1, x′

2, x′

3)

12. ਓήʔϜʹ͓͚Δަবू߹ͷྫɿίΞ͕ଘࡏ

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

ΛίΞʹଐ͞ͳ͍഑෼ͱ͠
͜Ε͕ෆ҆ఆͰ͋Δ͜ͱΛࣔ͢
ίΞʹଐ͞ͳ͍ͨΊ

͸

·ͨ͸

Λຬͨ͢

ͷͱ͖

Ͱ͋Δ

ΛͱΔ
ͨͩ͠

͸

Λຬͨ͢ਖ਼ͷ࣮਺


ΑΓ


͸
ϓϨΠϠʔͷϓϨΠϠʔʹର͢ΔҟٞʹͳΔ
·ͨ

Ͱ͋Δ͔Β
ϓϨΠϠʔ͸͜ͷҟٞʹର͢Δ‫ٯ‬ҟٞΛ΋ͨͳ͍Ώ͑ʹ

͸ෆ҆ఆ
(x′

1, x′

2, x′

3)
(x′

1, x′

2, x′

3) x′

1 + x′

2 10 x′

1 + x′

2 = 10, x′

2 3
x′

1 + x′

2 10 x′

3 0 (y′

1, y′

2, y′

3) = (x′

1 + ϵ, x′

2 + ϵ′

, ϵ′

′

)
ϵ, ϵ′

, ϵ′

′

ϵ + ϵ′

+ ϵ′

′

= 10 − (x′

1 + x′

2) 0
y′

1 + y′

2 = x′

1 + ϵ + x′

2 + ϵ′

v({1,2}) = 10
x′

1 x′

1 + ϵ = y′

1, x′

2 x′

2 + ϵ′

′

= y′

2
(y′

1, y′

2, y′

3) {1,2}
v({3}) = v({2, 3}) = 0
(x′

1, x′

2, x′

3)
‫ٯ‬ҟٞ



∑
k∈T
zk ≤ v(T)
zk ≥ xk, k ∈ T S
zk ≥ yk, k ∈ T ∩ S
ҟٞ


∑
k∈S
yk ≤ v(S)
yk xk, k ∈ S

(y′

1, y′

2, y′

3)

(x′

1, x′

2, x′

3)
ϓϨΠϠʔͷ

Λ·͔ͳ͑ͳ͍
x′

3

13. ਓήʔϜʹ͓͚Δަবू߹ͷྫɿίΞ͕ଘࡏ

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

ΛίΞʹଐ͞ͳ͍഑෼ͱ͠
͜Ε͕ෆ҆ఆͰ͋Δ͜ͱΛࣔ͢
ίΞʹଐ͞ͳ͍ͨΊ

͸

·ͨ͸

Λຬͨ͢


Ͱ͋Γ

ΛͱΔ
ͨͩ͠

͸

Λຬͨ͢ਖ਼ͷ࣮਺


ΑΓ


͸
ϓϨΠϠʔͷϓϨΠϠʔʹର͢ΔҟٞʹͳΔ
ˎϓϨΠϠʔͷϓϨΠϠʔʹର͢Δҟٞʹ͸‫ٯ‬ҟ͕ٞଘࡏ
·ͨ

Ͱ͋Δ͔Β
ϓϨΠϠʔ͸͜ͷҟٞʹର͢Δ‫ٯ‬ҟٞΛ΋ͨͳ͍Ώ͑ʹ

͸ෆ҆ఆ
(x′

1, x′

2, x′

3)
(x′

1, x′

2, x′

3) x′

1 + x′

2 10 x′

1 + x′

2 = 10, x′

2 3
x′

1 + x′

2 = 10, x′

2 3 x′

1 + x′

3 7 (y′

1, y′

2, y′

3) = (x′

1 + ϵ, 3, x′

3 + ϵ′

′

)
ϵ, ϵ′

′

ϵ + ϵ′

′

= 7 − (x′

1 + x′

3) 0
y′

1 + y′

3 = x′

1 + ϵ + x′

3 + ϵ′

′

v({1,3}) = 7
x′

1 x′

1 + ϵ = y′

1, x′

3 x′

3 + ϵ′

′

= y′

3
(y′

1, y′

2, y′

3) {1,3}
v({2}) = v({2,3}) = 0
(x′

1, x′

2, x′

3)

(y′

1, y′

2, y′

3)
‫ٯ‬ҟٞ



∑
k∈T
zk ≤ v(T)
zk ≥ xk, k ∈ T S
zk ≥ yk, k ∈ T ∩ S
ҟٞ


∑
k∈S
yk ≤ v(S)
yk xk, k ∈ S

(x′

1, x′

2, x′

3)

14. ਓήʔϜʹ͓͚Δަবू߹ͷྫɿίΞ͕ଘࡏ͠ͳ͍

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

ͷΈ͕ަবू߹ʹ‫·ؚ‬ΕΔ
(x1, x2, x3) =
(
10
3
,
10
3
,
10
3 )

(x1, x2, x3) =
(
10
3
,
10
3
,
10
3 )

15. ਓήʔϜʹ͓͚Δަবू߹ͷྫɿίΞ͕ଘࡏ͠ͳ͍

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

ͷΈ͕ަবू߹ʹ‫·ؚ‬ΕΔ
҆ఆੑͷ֬ೝ
഑෼

ΛͱΔ
ͨͩ͠

͸

Λຬͨ͢ਖ਼ͷ࣮਺


ϓϨΠϠʔͷϓϨΠϠʔʹରͯ͠ҟٞ

Λ΋ͭ
(x1, x2, x3) =
(
10
3
,
10
3
,
10
3 )
(y1, y2, y3) = (x1 + ϵ, ϵ′

, x3 + ϵ′

′

)
ϵ, ϵ′

, ϵ′

′

ϵ + ϵ′

+ ϵ′

′

= 10 − (x1 + x3)
y1 + y3 = x1 + ϵ + x3 + ϵ′

′

v({1,3}) = 10
x1 x1 + ϵ = y1, x3 x3 + ϵ′

′

= y3
((y1, y2, y3), {1,3})

(x1, x2, x3) =
(
10
3
,
10
3
,
10
3 )

(y1, y2, y3)
ҟٞ


∑
k∈S
yk ≤ v(S)
yk xk, k ∈ S

16. ਓήʔϜʹ͓͚Δަবू߹ͷྫɿίΞ͕ଘࡏ͠ͳ͍

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

ͷΈ͕ަবू߹ʹ‫·ؚ‬ΕΔ
҆ఆੑͷ֬ೝ
഑෼

ΛͱΔ
ϓϨΠϠʔͷϓϨΠϠʔʹରͯ͠ҟٞ

Λ΋ͭ
͔͠͠ϓϨΠϠʔ͸͜ͷҟٞʹରͯ͠‫ٯ‬ҟٞΛ΋ͭɿ

ͨͩ͠

͸

Λຬͨ͢ඇෛͷ࣮਺



Ώ͑ʹ

͸҆ఆ
(x1, x2, x3) =
(
10
3
,
10
3
,
10
3 )
(y1, y2, y3) = (x1 + ϵ, ϵ′

, x3 + ϵ′

′

)
((y1, y2, y3), {1,3})
(z1, z2, z3) = (ϵ̄, x2 + ϵ̄′

, y3 + ϵ̄′

′

)
ϵ̄, ϵ̄′

, ϵ̄′

′

ϵ + ϵ′

+ ϵ′

′

= 10 − (x2 + y3)
z2 + z3 = x2 + ϵ̄′

+ y3 + ϵ̄′

′

≤ v({2,3}) = 10
x2 ≤ x2 + ϵ̄′

= z2 y3 ≤ x3 + ϵ̄′

′

= z3
(x1, x2, x3) =
(
10
3
,
10
3
,
10
3 )
‫ٯ‬ҟٞ



∑
k∈T
zk ≤ v(T)
zk ≥ xk, k ∈ T S
zk ≥ yk, k ∈ T ∩ S

(y1, y2, y3)

(z1, z2, z3)

17. ਓήʔϜʹ͓͚Δަবू߹ͷྫɿίΞ͕ଘࡏ͠ͳ͍

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

ͷΈ͕ަবू߹ʹ‫·ؚ‬ΕΔ

Ҏ֎͕҆ఆͰͳ͍͜ͱΛࣔ͢
೚ҙͷ఺

ΛͱΓ
ҰൠੑΛࣦ͏͜ͱͳ͘

ͱ͢Δ

ͱͳΔਖ਼ͷ࣮਺

ΛͱΓ
഑෼

ΛͱΔ


Ώ͑ʹ


ΑΓϓϨΠϠʔͷϓϨΠϠʔʹରͯ͠ҟٞ

Λ΋ͭ
(x1, x2, x3) =
(
10
3
,
10
3
,
10
3 )
(x1, x2, x3) =
(
10
3
,
10
3
,
10
3 )
(x′

1, x′

2, x′

3) ≠ (x1, x2, x3)
x′

1 ≥ x′

2 ≥ x′

3, x′

1 x′

3
ϵ
x′

1 − x′

3
2
ϵ (y1, y2, y3) = (ϵ,10 − (x′

3 + 2ϵ), x′

3 + ϵ)
y2 + y3 = 10 − (x′

3 + 2ϵ) + x′

3 + ϵ′

′

v({2,3}) = 10
x′

2 − y2 = x′

2 − {10 − (x′

3 + 2ϵ)} − 10 + x′

1 + x′

2 ≤ 0, x′

2 y2
x′

3 x′

3 + ϵ = y3
((y1, y2, y3), {2,3})

(z1, z2, z3)

(y1, y2, y3)

(x1, x2, x3)
ҟٞ


∑
k∈S
yk ≤ v(S)
yk xk, k ∈ S

18. ਓήʔϜʹ͓͚Δަবू߹ͷྫɿίΞ͕ଘࡏ͠ͳ͍

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
೚ҙͷ఺

ΛͱΓ
ҰൠੑΛࣦ͏͜ͱͳ͘

ͱ͢Δ

ͱͳΔਖ਼ͷ࣮਺

ΛͱΓ
഑෼

ΛͱΔ
ΑΓϓϨΠϠʔͷϓϨΠϠʔʹରͯ͠ҟٞ

Λ΋ͭ
͜ͷҟٞʹରͯ͠ϓϨΠϠʔ͸‫ٯ‬ҟٞΛ΋ͨͳ͍ɿ

Ͱ͋Δ͔Β
ਓఏ‫ʹܞ‬ΑΔ‫ٯ‬ҟٞ͸ͳ͍
‫ٯ‬ҟٞ

͕͋Δͱ͢Δͱ

·ͨ


Ͱͳ͚Ε͹͍͚ͳ͍͕
‫ͭ̎ऀޙ‬ͷෆ౳ࣜͱ

ͷऔΓํ͔Β


ͱͳΓ
ໃ६
͕ͨͬͯ͠ϓϨΠϠʔ͸‫ٯ‬ҟٞΛ΋ͨͳ͍
(x′

1, x′

2, x′

3) ≠ (x1, x2, x3)
x′

1 ≥ x′

2 ≥ x′

3, x′

1 x′

3
ϵ
x′

1 − x′

3
2
ϵ (y1, y2, y3) = (ϵ,10 − (x′

3 + 2ϵ), x′

3 + ϵ)
((y1, y2, y3), {2,3})
v({1}) = 0
((z1, z2, z3), {1,2})
z1 + z2 ≤ v({1,2}) = 10 z1 ≥ x′

1 z2 ≥ y2 = 10 − (x′

3 + 2ϵ)
ϵ
z1 + z2 ≥ x′

1 + 10 − (x′

3 + 2ϵ) 10 = v({1,2}) ≥ z1 + z2

(z1, z2, z3)

(y1, y2, y3)

(x1, x2, x3)
‫ٯ‬ҟٞ



∑
k∈T
zk ≤ v(T)
zk ≥ xk, k ∈ T S
zk ≥ yk, k ∈ T ∩ S

19. ήʔϜཧ࿦#4*$ୈճ
ଓɾަবू߹
࣍ճɿΧʔωϧ