https://youtu.be/T7jA3ZK2kmQ

1. ήʔϜཧ࿦#4*$ୈճ
ਔʹؔ͢Δఆཧͷূ໌

2. ఏ‫ܞ‬ͷෆຬ ෮श
഑෼͕ड༰తͰ͋Δ͜ͱͷఆٛ ෮श
ਔͷఆٛ ෮श
ਔ͸ඞͣଘࡏ͢Δ

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·ؚ‬ΕΔਔ

Χʔωϧ

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

6. ਔ͸ඞͣଘࡏ͢Δ
ఆཧ
ήʔϜ

ʹ͓͍ͯ
ਔΛ

ͱ͢Δͱ

Ͱ͋Δ
ূ໌
֤഑෼

ʹ͍ͭͯ
ෆຬϕΫτϧ

ʹؔͯ͠

͸

ʹ͓͚Δ֤ఏ‫ܞ‬

ͷෆຬͷྔ

Λ
େ͖ͳ΋ͷ͔Βॱʹฒ΂ͨϕΫτϧͳͷͰ

ͱදͤΔ
(N, v)
𝒩
(v)
𝒩
(v) ≠ ∅
x ∈
𝒜
(v) θ(x) = (θ1(x), ⋯, θ2n−2(x))
θi(x), i = 1,⋯,2n
− 2 x S, S ≠ N, ∅ e(S, x)
θi(x) = max
R⊆2N
{N,∅},|R|=i
(min
S∈R
e(S, x))

7. ਔ͸ඞͣଘࡏ͢Δ








ͱ͢Δͱ













θi(x) = max
R⊆2N
{N,∅},|R|=i
(min
S∈R
e(S, x))
N = {1,2,3} 2N
= {∅, {1}, {2}, {3} {1,2}, {1,3}, {2,3} {1,2,3}}
v(∅) = v({1}) = v({2}) = v({2}) = 0 v({1,2}) = 5, v({1,3}) = 7, v({2,3}) = 8, v({1,2,3}) = 10
x = (1,3,6)
e({1}, x) = − 1 e({2}, x) = − 3 e({3}, x) = − 6 e({1,2}, x) = 1 e({1,3}, x) = 0 e({2,3}, x) = − 1
θ(x) = (1,0, − 1, − 1, − 3, − 6)
θ1(x) = max
R⊆2N
{N,∅},|R|=1
(min
S∈R
e(S, x)) max
R∈{{{1}},{{2}},{{3}},{{1,2}},{{1,3}},{{2,3}}}
(min
S∈R
e(S, x))
max{e({1}, x), e({2}, x), e({3}, x), e({1,2}, x), e({1,3}, x), e({2,3}, x)} = 1
θ2(x) = max
R⊆2N
{N,∅},|R|=2
(min
S∈R
e(S, x)) max
R∈{{{1},{2}},{{1},{3}},⋯,{{1,2},{1,3}}},⋯{{1,3},{2,3}}}
(min
S∈R
e(S, x))
max{−3, − 6,⋯,0,⋯, − 1} = 0 ̎ͭͷू߹Λൺֱͯ͠খ͍͞ΛબͿ ൪େ͖͍ෆຬ͕আ͔ΕΔ

8. 












θi(x) = max
R⊆2N
{N,∅},|R|=i
(min
S∈R
e(S, x))
θ3(x) = max
R⊆2N
{N,∅},|R|=3
(min
S∈R
e(S, x)) max
R∈{{{1},{2},{3}},⋯,{{1,2},{1,3},{2,3}}}
(min
S∈R
e(S, x))
max{−6,⋯, − 1} = − 1
θ4(x) = max
R⊆2N
{N,∅},|R|=4
(min
S∈R
e(S, x)) max
R∈{{{1},{2},{3},{1,2}},⋯,{{3},{1,2},{1,3},{2,3}}}
(min
S∈R
e(S, x))
max{−6,⋯, − 6} = − 1
θ5(x) = max
R⊆2N
{N,∅},|R|=5
(min
S∈R
e(S, x)) max
R∈{{{1},{2},{3},{1,2},{1,3}},⋯,{{2},{3},{1,2},{1,3},{2,3}}}
(min
S∈R
e(S, x))
max{−6,⋯, − 6} = − 3
θ6(x) = max
R⊆2N
{N,∅},|R|=6
(min
S∈R
e(S, x)) max
R∈{{{1},{2},{3},{1,2},{1,3},{2,3}}}
(min
S∈R
e(S, x))
max{−6} = − 6
ͭͷू߹Λൺֱͯ͠খ͍͞ΛબͿ ൪໨ͱ൪໨ʹେ͖͍ෆຬ͕আ͔ΕΔ

9. 
ʹ͍ͭͯ
ෆຬϕΫτϧ

ʹؔͯ͠

͸

ʹ͓͚Δ֤ఏ‫ܞ‬

ͷෆຬͷྔ

Λ
େ͖ͳ΋ͷ͔Βॱʹฒ΂ͨϕΫτϧͳͷͰ

ͱදͤΔ

͸

ʹؔͯ͠࿈ଓؔ਺Ͱ͋Γ

΋࿈ଓؔ਺
x ∈
𝒜
(v) θ(x) = (θ1(x), ⋯, θ2n−2(x))
θi(x), i = 1,⋯,2n
− 2 x S, S ≠ N, ∅ e(S, x)
θi(x) = max
R⊆2N
{N,∅},|R|=i
(min
S∈R
e(S, x))
e(S, x) = v(S) −
∑
i∈S
xi x θi(x)

10. ਔ͸ඞͣଘࡏ͢Δ
ূ໌



ͱ͢Δ
഑෼ͷू߹

͸ίϯύΫτͰ

͸࿈ଓؔ਺Ώ͑
࠷খ஋͕ଘࡏ͠

͔ͭίϯύΫτ
ίϯύΫτू߹্

Ͱఆٛ͞Ε࣮ͨ਺஋ؔ਺

͕࿈ଓͱ͢Ε͹

͸

্Ͱ࠷େ஋ɾ࠷খ஋Λ΋ͭ
θi(x) = max
R⊆2N
{N,∅},|R|=i
(min
S∈R
e(S, x))
A1
= {x ∈
𝒜
(v)|θ1(x) = min
y∈
𝒜
(v)
θ1(y)}
Ak
= {x ∈ Ak−1
|θk(x) = min
y∈Ak−1
θk(y)}, k = 2,⋯,2n
− 2
𝒜
(v) θ1(x) A1
≠ ∅
K ⊆ ℜn
f : K → ℜ
f K

11. ਔ͸ඞͣଘࡏ͢Δ
ূ໌



ͱ͢Δ
഑෼ͷू߹

͸ίϯύΫτͰ

͸࿈ଓؔ਺Ώ͑
࠷খ஋͕ଘࡏ͠

͔ͭίϯύΫτ
഑෼ͷू߹

͸ίϯύΫτͰ

͸࿈ଓؔ਺Ώ͑
࠷খ஋͕ଘࡏ͠

͔ͭίϯύΫτ
ɾɾɾ
഑෼ͷू߹

͸ίϯύΫτͰ

͸࿈ଓؔ਺Ώ͑
࠷খ஋͕ଘࡏ͠


͸ਔͰ͋Δʢূ໌ऴྃ
θi(x) = max
R⊆2N
{N,∅},|R|=i
(min
S∈R
e(S, x))
A1
= {x ∈
𝒜
(v)|θ1(x) = min
y∈
𝒜
(v)
θ1(y)}
Ak
= {x ∈ Ak−1
|θk(x) = min
y∈Ak−1
θk(y)}, k = 2,⋯,2n
− 2
𝒜
(v) θ1(x) A1
≠ ∅
A1
θ2(x) A2
≠ ∅
A2n
−3
θ2n−2(x) A2n
−2
≠ ∅
A2n
−2

12. 
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
ਔ͸ඞͣଘࡏ͢Δ
0
M
−5
−
10
3
−1
−
3
2

v({1,2,3}) = v({1,2}) = 10, v({1,3}) = 7, v({2,3}) = 0, v({1}) = v({2}) = v({3}) = 0

13. ਔ͸ඞͣଘࡏ͢Δ
࠷େෆຬΛ࠷খɿ







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



0 ≤ x1 ≤ 10
0 ≤ x2 ≤ 3
x3 = 0

v({1,2,3}) = v({1,2}) = 10, v({1,3}) = 7, v({2,3}) = 0, v({1}) = v({2}) = v({3}) = 0
A1 = A2 = {x ∈
𝒜
(v)|
}

14. 
(θ1(x), θ2(x), θ3(x), θ4(x), θ5(x), θ6(x)) = (0, 0, θ3(x), θ4(x), θ5(x), θ6(x))
ਔ͸ඞͣଘࡏ͢Δ

v({1,2,3}) = v({1,2}) = 10, v({1,3}) = 7, v({2,3}) = 0, v({1}) = v({2}) = v({3}) = 0
1
2 3
x1 = 10
x2 = 3
x3 = 0
𝒜
(v)
A1 = A2

15. ਔ͸ඞͣଘࡏ͢Δ

v({1,2,3}) = v({1,2}) = 10, v({1,3}) = 7, v({2,3}) = 0, v({1}) = v({2}) = v({3}) = 0
൪໨ʹେ͖͍ෆຬΛ࠷খɿ





M′

= −
3
2
e({1,3}, x) = x2 − 3 = −
3
2
e({2,3}, x) = x1 − 10 = −
3
2
e({1}, x) = − x1 = −
17
2
e({2}, x) = − x2 = −
3
2

16. 
(θ1(x), θ2(x), θ3(x), θ4(x), θ5(x), θ6(x)) = (0, 0, −
3
2
, −
3
2
, −
3
2
, −
17
2
)
ਔ͸ඞͣଘࡏ͢Δ

v({1,2,3}) = v({1,2}) = 10, v({1,3}) = 7, v({2,3}) = 0, v({1}) = v({2}) = v({3}) = 0
1
2 3
𝒜
(v)
x1 = 10
x2 = 3
x3 = 0
A1 = A2

17. ήʔϜཧ࿦#4*$ୈճ
ਔʹؔ͢Δఆཧͷূ໌
࣍ճɿਔʹؔ͢Δఆཧͷূ໌