https://www.nicovideo.jp/watch/sm37233994 https://www.youtube.com/watch?v=d3Ze6tuA1Qc&t=301s https://www.youtube.com/channel/UCEr2VZ6DI4gATRtVJw49ZFw

2.
S e(S, x) = v(S) −
∑
i∈S
xi
x, x′ k ( 1 ≤ k ≤ 2n
− 4)
θl(x) = θl(x′) ∀l = 1,2,⋯k − 1, θk(x) < θk(x′)
x x′ x >L x′
x x
x
𝒩(v) = {x ∈ 𝒜(v)|x′ >L x x′ }

3.
S e(S, x) = v(S) −
∑
i∈S
xi
x, x′ k ( 1 ≤ k ≤ 2n
− 4)
θl(x) = θl(x′) ∀l = 1,2,⋯k − 1, θk(x) < θk(x′)
x x′ x >L x′
x x
x
𝒩(v) = {x ∈ 𝒜(v)|x′ >L x x′ }
min M
e({3}, x) ≤ M
s . t .
x1 + x2 + x3 = ν(N)
x1 ≥ ν({1}), x2 ≥ ν({2}), x3 ≥ ν({3})
e({1,2}, x) ≤ M
e({2,3}, x) ≤ M
e({1,3}, x) ≤ M
e({1}, x) ≤ M
e({2}, x) ≤ M

4.
ν({A}) = ν({B}) = ν({C}) = 0
ν({A, B}) = 500, ν({A, C}) = 1000, ν({B, C}) = 0
ν({A, B, C}) = 1000

5.
ν({A}) = ν({B}) = ν({C}) = 0
ν({A, B}) = 500, ν({A, C}) = 1000, ν({B, C}) = 0
ν({A, B, C}) = 1000
min M
e({C}, x) = 0 − xC ≤ M
s . t .
xA + xB + xC = 1000
xA, xB, xC ≥ 0
e({A, B}, x) = 500 − (xA + xB) = − 500 + xC ≤ M
e({A, C}, x) = 1000 − (xA + xC) = xB ≤ M
e({B, C}, x) = 0 − (xB + xC) = − 1000 + xA ≤ M
e({A}, x) = 0 − xA ≤ M
e({B}, x) = 0 − xB ≤ M

6.
ν({A}) = ν({B}) = ν({C}) = 0
ν({A, B}) = 500, ν({A, C}) = 1000, ν({B, C}) = 0
ν({A, B, C}) = 1000
min M
s . t .
−3M ≤ xA + xB + xC = 1000 ≤ 3M + 1500
−M ≤ xC ≤ M + 500
−M ≤ xB ≤ M
−M ≤ xA ≤ M + 1000
0 ≤ xC ≤ M + 500
0 ≤ xB ≤ M
0 ≤ xA ≤ M + 1000

7.
ν({A}) = ν({B}) = ν({C}) = 0
ν({A, B}) = 500, ν({A, C}) = 1000, ν({B, C}) = 0
ν({A, B, C}) = 1000
min M
s . t .
M ≥ −
1000
3
, M ≥ −
500
3
−M ≤ xC ≤ M + 500
−M ≤ xB ≤ M
−M ≤ xA ≤ M + 1000
0 ≤ xC ≤ M + 500
0 ≤ xB ≤ M
0 ≤ xA ≤ M + 1000
M
xA, xB, xC
−
500
3
−
1000
3

8.
ν({A}) = ν({B}) = ν({C}) = 0
ν({A, B}) = 500, ν({A, C}) = 1000, ν({B, C}) = 0
ν({A, B, C}) = 1000
min M
s . t .
M
xA, xB, xC
−
500
3
M ≥ −
1000
3
, M ≥ −
500
3
−M ≤ xC ≤ M + 500
−M ≤ xB ≤ M
−M ≤ xA ≤ M + 1000
0 ≤ xC ≤ M + 500
0 ≤ xB ≤ M
0 ≤ xA ≤ M + 1000
−
1000
3

9.
ν({A}) = ν({B}) = ν({C}) = 0
ν({A, B}) = 500, ν({A, C}) = 1000, ν({B, C}) = 0
ν({A, B, C}) = 1000
0 ≤ xC ≤ 500
0 ≤ xB ≤ 0 ⇒ xB = 0
0 ≤ xA ≤ 1000
xA + xC = 1000 M
xA, xB, xC
−
500
3
−
1000
3

10.
ν({A}) = ν({B}) = ν({C}) = 0
ν({A, B}) = 500, ν({A, C}) = 1000, ν({B, C}) = 0
ν({A, B, C}) = 1000
min M′
s . t .
xB = 0
e({C}, x) = 0 − xC ≤ M′
xA + xC = 1000
xA, xC ≥ 0
e({A, B}, x) = 500 − (xA + 0) = − 500 + xC ≤ M′
(e({A, C}, x) = 1000 − (xA + xC) = 0 = M)
e({B, C}, x) = 0 − (0 + xC) = − 1000 + xA ≤ M′
e({A}, x) = 0 − xA ≤ M′
(e({B}, x) = 0 − 0 = 0 = M)

11.
ν({A}) = ν({B}) = ν({C}) = 0
ν({A, B}) = 500, ν({A, C}) = 1000, ν({B, C}) = 0
ν({A, B, C}) = 1000
min M′
s . t .
xB = 0
−2M ≤ xA + xC = 1000 ≤ 2M′+ 1500
0 ≤ xC ≤ M′+ 500
0 ≤ xA ≤ M′+ 1000
−M′ ≤ xC ≤ M′+ 500
−M′ ≤ xA ≤ M′+ 1000

12.
ν({A}) = ν({B}) = ν({C}) = 0
ν({A, B}) = 500, ν({A, C}) = 1000, ν({B, C}) = 0
ν({A, B, C}) = 1000
min M′
s . t .
xB = 0
M
xA, xC
−250
−250
0 ≤ xC ≤ M′+ 500
0 ≤ xA ≤ M′+ 1000
−M′ ≤ xC ≤ M′+ 500
−M′ ≤ xA ≤ M′+ 1000
−2M ≤ xA + xC = 1000 ≤ 2M′+ 1500

13.
ν({A}) = ν({B}) = ν({C}) = 0
ν({A, B}) = 500, ν({A, C}) = 1000, ν({B, C}) = 0
ν({A, B, C}) = 1000
xA + xC = 1000
250 ≤ xC ≤ 250
250 ≤ xA ≤ 750
⇒ xA = 750, xC = 250
(750, 0, 250)
M
xA, xC
−250
−250

14.
ν({A}) = ν({B}) = ν({C}) = 0
ν({A, B}) = 500, ν({A, C}) = 1000, ν({B, C}) = 0
ν({A, B, C}) = 1000
Core(ν) = {x|xA + xC = 1000, xB = 0, 500 ≤ xA ≤ 1000 (, 0 ≤ xC ≤ 500)}xC
xA
xC ≤ 500
xA ≤ 1000
xB = 0
(750, 0, 250)

16.
ν({A}) = 50, ν({B}) = ν({C}) = 0
ν({A, B}) = 250, ν({A, C}) = 250, ν({B, C}) = 150
ν({A, B, C}) = 300

17.
xA + xB + xc = ν({A, B, C}) = 300
x = (xA, xB, xC)
xA ≥ ν({A}) = 50, xB ≥ ν({B}) = 0, xC ≥ ν({C}) = 0
xA + xB ≥ ν({A, B}) = 250
xA + xC ≥ ν({A, C}) = 250
xB + xC ≥ ν({B, C}) = 150
ν({A}) = 50, ν({B}) = ν({C}) = 0
ν({A, B}) = 250, ν({A, C}) = 250, ν({B, C}) = 150
ν({A, B, C}) = 300

18.
xA + xB + xc = ν({A, B, C}) = 300
x = (xA, xB, xC)
xA ≥ ν({A}) = 50, xB ≥ ν({B}) = 0, xC ≥ ν({C}) = 0
xA + xB ≥ ν({A, B}) = 250 ⇒ 50 ≥ xC
xA + xC ≥ ν({A, C}) = 250 ⇒ 50 ≥ xB
xB + xC ≥ ν({B, C}) = 150 ⇒ 150 ≥ xA
ν({A}) = 50, ν({B}) = ν({C}) = 0
ν({A, B}) = 250, ν({A, C}) = 250, ν({B, C}) = 150
ν({A, B, C}) = 300
250 ≥ xA + xB + xC = 300

19.
x = (xA, xB, xC)
ν({A}) = 50, ν({B}) = ν({C}) = 0
ν({A, B}) = 250, ν({A, C}) = 250, ν({B, C}) = 150
ν({A, B, C}) = 300
min M
e({C}, x) = 0 − xC ≤ M
s . t .
xA + xB + xC = 300
xA ≥ 50, xB, xC ≥ 0
e({A, B}, x) = 250 − (xA + xB) = − 50 + xC ≤ M
e({A, C}, x) = 250 − (xA + xC) = − 50 + xB ≤ M
e({B, C}, x) = 150 − (xB + xC) = − 150 + xA ≤ M
e({A}, x) = 50 − xA ≤ M
e({B}, x) = 0 − xB ≤ M

20.
x = (xA, xB, xC)
ν({A}) = 50, ν({B}) = ν({C}) = 0
ν({A, B}) = 250, ν({A, C}) = 250, ν({B, C}) = 150
ν({A, B, C}) = 300
min M
−M ≤ xC ≤ M + 50
s . t .
50 − 3M ≤ xA + xB + xC = 300 ≤ 3M + 250
0 ≤ xC ≤ M + 50
0 ≤ xB ≤ M + 50
50 ≤ xA ≤ M + 150
50 − M ≤ xA ≤ M + 150
−M ≤ xB ≤ M + 50

21.
ν({A}) = 50, ν({B}) = ν({C}) = 0
ν({A, B}) = 250, ν({A, C}) = 250, ν({B, C}) = 150
ν({A, B, C}) = 300
min M
−M ≤ xC ≤ M + 50
s . t .
50 − 3M ≤ xA + xB + xC = 300 ≤ 3M + 250
0 ≤ xC ≤ M + 50
0 ≤ xB ≤ M + 50
50 ≤ xA ≤ M + 150
50 − M ≤ xA ≤ M + 150
−M ≤ xB ≤ M + 50
M
xA, xB, xC
50
3
50
3
−
250
3

22.
ν({A}) = 50, ν({B}) = ν({C}) = 0
ν({A, B}) = 250, ν({A, C}) = 250, ν({B, C}) = 150
ν({A, B, C}) = 300
xA + xB + xC = 300
0 ≤ xC ≤
50
3
+ 50
0 ≤ xB ≤
50
3
+ 50
50 ≤ xA ≤
50
3
+ 150 M
xA, xB, xC
50
3
50
3
−
250
3
(
50
3
+ 150,
50
3
+ 50,
50
3
+ 50) = (
500
3
,
200
3
,
200
3
)

23.
ν({A}) = 50, ν({B}) = ν({C}) = 0
ν({A, B}) = 250, ν({A, C}) = 250, ν({B, C}) = 150
ν({A, B, C}) = 300
xA + xB + xC = 300
0 ≤ xC ≤
50
3
+ 50
0 ≤ xB ≤
50
3
+ 50
50 ≤ xA ≤
50
3
+ 150 M
xA, xB, xC
50
3
50
3
−
250
3
(
50
3
+ 150,
50
3
+ 50,
50
3
+ 50) = (
500
3
,
200
3
,
200
3
)

24. xC = 50
xA = 150
xB = 50
ν({A}) = 50, ν({B}) = ν({C}) = 0
ν({A, B}) = 250, ν({A, C}) = 250, ν({B, C}) = 150
ν({A, B, C}) = 300
(
50
3
+ 150,
50
3
+ 50,
50
3
+ 50) = (
500
3
,
200
3
,
200
3
)
xA = 50

25. See you next time