https://youtu.be/uPqxcTwxc2A

1. ήʔϜཧ࿦#4*$ୈճ
ަবू߹

2. ҟٞ
‫ٯ‬ҟٞ
ަবू߹
ίΞͱަবू߹
ਓήʔϜʹ͓͚Δަবू߹ͷྫ

3. ҟٞ
ҟٞ͋Γ
ަবΛҟٞɾ‫ٯ‬ҟٞͱ͍͏‫Ͱܗ‬ఆࣜԽ

4. ҟٞ
ҟٞ͋Γ
ަবΛҟٞɾ‫ٯ‬ҟٞͱ͍͏‫Ͱܗ‬ఆࣜԽ

5. ҟٞ
ҟٞ͋Γ
ަ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)
࣮‫ݱ‬Մೳੑ

6. ‫ٯ‬ҟٞ
ҟٞ͋Γ
ަবΛҟٞɾ‫ٯ‬ҟٞͱ͍͏‫Ͱܗ‬ఆࣜԽ

7. ‫ٯ‬ҟٞ
ҟٞ͋Γ
ަ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)

8. ަ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)

9. ίΞͱަবू߹
ఆཧɿίΞͱަবू߹ͷؔ܎
ήʔϜ

ʹ͓͍ͯ
ίΞΛ

ަবू߹Λ

ͱ͢Δͱ

ˎަবू߹͸ৗʹඇۭͰ͋Δ ผ‫ػ‬ձʹূ໌
ূ໌
ίΞ

Ͱ͋Ε͹໌Β͔

ͱ͢Δ

Ͱ͸ͳ͍ͱԾఆ͠

ΛͱΔ

Ͱ͋ΔͨΊ
͋ΔϓϨΠϠʔ

͔Β

ʹର͢Δҟ͕ٞଘࡏ͢Δ͢ͳΘͪ
͋Δఏ‫ܞ‬



ͱ


ͱͳΔ

͕ଘࡏ͢Δ͜ͷͱ͖

ͱͳΓ

ʹໃ६͢Δ
(N, v) C(v) ℬ(v) C(v) ⊆ ℬ(v)
C(v) = ∅ C(v) ≠ ∅ C(v) ⊆ ℬ(v) x ∈ C(v)ℬ(v)
x ∉ ℬ(v) i j
S i ∈ S j ∉ S
∑
k∈S
yk ≤ v(S) yk xk, k ∈ S y
∑
k∈S
xk
∑
k∈S
yk ≤ v(S)
x ∈ C(v)

10. 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)}

11. 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, x2, x3)

12. 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

(x1, x2, x3)

13. 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)

14. ήʔϜཧ࿦#4*$ୈճ
ަবू߹
࣍ճɿଓɾަবू߹