https://youtu.be/vQ3lpwaToZk

1. ήʔϜཧ࿦#4*$ୈճ
Χʔωϧ

2. Χʔωϧ֓ཁ
ఏ‫ܞ‬ͷෆຬͱϓϨΠϠʔͷ༏Ґੑ
Χʔωϧͷఆٛ
ਓήʔϜʹ͓͚ΔΧʔωϧͷྫ

3. Χʔωϧ֓ཁ
༨ஊ
Χʔωϧ͸ίϯϐϡʔλʹ͓͚Δ04಺ʹ͋Δॏཁͳ‫ػ‬ೳΛ΋ͭ
ॏཁͳ
ͭ·Γ֩ͱͳΔ‫ػ‬ೳΛ୲͏
Χʔωϧ LFSOFM
͸֩ͱ༁͞ΕΔ
ਔ OVDMFPMVT
΋೔ຊ‫ޠ‬తʹ͸͕֩ͩ
த৺ͷதͷத৺ ʁ
ͱ͍͏Πϝʔδ
ਔ

Χʔωϧ

ަবू߹
⊆ ⊆
ަবू߹
Χʔωϧ
ਔ

4. Χʔωϧ֓ཁ
֤ϓϨΠϠʔ͝ͱͷަবΛ
ෆຬͱ͍͏‫Ͱܗ‬ଊ͑Δ
഑෼ʹෆຬɿෆຬ ഑෼ʹෆຬɿෆຬ
ӈͷϓϨΠϠʔ͸େ͖͍ෆຬΛ΋ͪ
΋ͬͱେ͖͍རಘΛཁ‫͏͠ٻ‬Δͱ͍͏ෆ҆ఆͳঢ়ଶ

5. ఏ‫ܞ‬ͷෆຬͱϓϨΠϠʔͷ༏Ґੑ
ఆٛɿఏ‫ܞ‬

ͷෆຬ
ಛੑؔ਺‫ܗ‬ήʔϜ

ʹ͓͍ͯ
഑෼

ͱఏ‫ܞ‬

ΛͱΔͱ͖

Λ഑෼

ʹର͢Δఏ‫ܞ‬

ͷෆຬͱ͍͏
ਓͷϓϨΠϠʔ


ΛͱΓ

ͱ͓͘
ఆٛɿ࠷େෆຬ
ਓͷϓϨΠϠʔ


ͱ഑෼

ʹ͍ͭͯ

Λ഑෼

ʹ͓͚ΔϓϨΠϠʔ

ͷ

ʹର͢Δ࠷େෆຬͱ͍͏
S
(N, v) x ∈
𝒜
(v) S ⊆ N
e(S, x) = v(S) −
∑
i∈S
xi x S
i j ∈ N Tij = {S ⊆ N|i ∈ S, j ∉ S}
i j ∈ N x ∈
𝒜
(v)
sij(x) = max
S∈Tij
e(S, x)
x i j

6. ఏ‫ܞ‬ͷෆຬͱϓϨΠϠʔͷ༏Ґੑ
ఆٛɿ༏Ґ
ਓͷϓϨΠϠʔ


ͱ഑෼

ʹ͍ͭͯ


ͱͳΔͱ͖
഑෼

ʹ͓͍ͯϓϨΠϠʔ

͸

ΑΓ΋༏ҐͰ͋Δͱ͍͍

ͱॻ͘

Ͱ͋Ε͹
ϓϨΠϠʔ

͸

ʹରͯ͠རಘΛ͘Εͱ͸͍͑ͳ͍
ͳͥͳΒ

͸ఏ‫ܞ‬Λ૊ΉΠϯηϯςΟϒ͕ͳ͘ͳΔͨΊ
i j ∈ N x ∈
𝒜
(v)
sij(x) sji(x) xj v({j})
x i j i ≻x j
xj ≤ v({j}) i j
j
ਓͷϓϨΠϠʔͷ
࠷େෆຬΛൺֱ

7. Χʔωϧͷఆٛ
ఆٛɿ‫ߧۉ‬ঢ়ଶ
ਓͷϓϨΠϠʔ


ͱ഑෼

ʹ͍ͭͯ

Ͱ΋ͳ͘

Ͱ΋ͳ͍ͱ͖
഑෼

ʹ͓͍ͯϓϨΠϠʔ

ͱ

͸‫ߧۉ‬ঢ়ଶͰ͋Δͱ͍͍

ͱॻ͘
ఆٛɿΧʔωϧ
೚ҙͷਓͷϓϨΠϠʔ͕‫ߧۉ‬ঢ়ଶʹ͋ΔΑ͏ͳ഑෼ͷશମ

ΛΧʔωϧͱ͍͏
i j ∈ N x ∈
𝒜
(v)
i ≻x j j ≻x i
x i j i ∼x j
𝒦
= {x ∈
𝒜
(v)|i ∼x j ∀i, j ∈ N, i ≠ j}

8. Χʔωϧͷఆٛ
ఆٛɿ‫ߧۉ‬ঢ়ଶ
ਓͷϓϨΠϠʔ


ͱ഑෼

ʹ͍ͭͯ

Ͱ΋ͳ͘

Ͱ΋ͳ͍ͱ͖
഑෼

ʹ͓͍ͯϓϨΠϠʔ

ͱ

͸‫ߧۉ‬ঢ়ଶͰ͋Δͱ͍͍

ͱॻ͘

ͱͳΔͨΊʹ͸

͸഑෼ͷ‫ݸ‬ਓ߹ཧੑΑΓ


ʹ஫ҙͯ͠

͔ͭ

͢ͳΘͪ

·ͨ͸

ͳΒ͹

·ͨ͸

ͳΒ͹

i j ∈ N x ∈
𝒜
(v)
i ≻x j j ≻x i
x i j i ∼x j
i ∼x j x ∈
𝒜
(v) xi ≥ v({i}) xj ≥ v({j})
sij(x) ≥ sji(x) sij(x) ≤ sji(x) sij(x) = sji(x)
sij(x) sji(x) xj = v({j})
sij(x) sji(x) xi = v({i})
ఆٛɿ༏Ґ
ਓͷϓϨΠϠʔ


ͱ഑෼

ʹ͍ͭͯ


ͱͳΔͱ͖
഑෼

ʹ͓͍ͯϓϨΠϠʔ

͸

ΑΓ΋༏ҐͰ͋Δͱ͍͍

ͱॻ͘
i j ∈ N x ∈
𝒜
(v)
sij(x) sji(x) xj v({j})
x i j i ≻x j

͔ͭ

xj ≥ v({j}) xj ≤ v({j})

͔ͭ

xi ≥ v({i}) xi ≤ v({i})

9. ਓήʔϜʹ͓͚ΔΧʔωϧͷྫ
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)}

10. ਓήʔϜʹ͓͚ΔΧʔωϧͷྫ

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
֤ϓϨΠϠʔͷ࠷େෆຬ͸





x = (x1, x2, x3) ∈
𝒜
(v)
s12(x) = max(v({1,3}) − x1 − x3), v({1}) − x1) )
= max(10 − x1 − x3, 0 − x1 )
= 10 − x1 − x3
s13(x) = max(v({1,2}) − x1 − x2, v({1}) − x1 )
= 10 − x1 − x2

10 − x1 − x3 ≥ 0 − x1

11. ਓήʔϜʹ͓͚ΔΧʔωϧͷྫ

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
֤ϓϨΠϠʔͷ࠷େෆຬ͸







x = (x1, x2, x3) ∈
𝒜
(v)
s12(x) = 10 − x1 − x3
s13(x) = 10 − x1 − x2
s21(x) = max(v({2,3}) − x2 − x3, v({2}) − x2 )
= max(0 − x2 − x3, 0 − x2 )
= − x2
s23(x) = max(v({1,2}) − x1 − x2, v({2}) − x2 )
= 10 − x1 − x2

12. ਓήʔϜʹ͓͚ΔΧʔωϧͷྫ

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
֤ϓϨΠϠʔͷ࠷େෆຬ͸






x = (x1, x2, x3) ∈
𝒜
(v)
s12(x) = 10 − x1 − x3
s13(x) = 10 − x1 − x2
s21(x) = − x2
s23(x) = 10 − x1 − x2
s31(x) = − x3
s32(x) = 10 − x1 − x3

ͱͳΔͨΊʹ͸

ͱͳΓໃ६

ͱͳΓ

͕഑෼Ͱ͋Δ͔Β
͜ͷ৔߹͸ଘࡏ͠ͳ͍

1 ∼x 2
s12(x) s21(x) ⇔ 10 − x1 − x3 − x2 ⇔ x2 0 = v({2})
s12(x) s21(x) ⇔ 10 − x1 − x3 − x2 ⇔ x2 0
x2
s12(x) = s21(x) ⇔ 10 − x1 − x3 = − x2 ⇔ x2 = 0

ͱͳΔͨΊʹ͸

·ͨ͸

ͳΒ͹

·ͨ͸

ͳΒ͹

i ∼x j
sij(x) = sji(x)
sij(x) sji(x) xj = v({j})
sij(x) sji(x) xi = v({i})

13. ਓήʔϜʹ͓͚ΔΧʔωϧͷྫ

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
֤ϓϨΠϠʔͷ࠷େෆຬ͸






x = (x1, x2, x3) ∈
𝒜
(v)
s12(x) = 10 − x1 − x3
s13(x) = 10 − x1 − x2
s21(x) = − x2
s23(x) = 10 − x1 − x2
s31(x) = − x3
s32(x) = 10 − x1 − x3

ͱͳΔͨΊʹ͸ɾɾɾ


ͱͳΔͨΊʹ͸ɾɾɾ


ͱͳΔͨΊʹ͸

ͳΒ͹

Ͱ͋Δ͕͜ͷͱ͖

ͱͳΓໃ६

ͳΒ͹

Ͱ͋Δ͕͜ͷͱ͖

ͱͳΓໃ६

1 ∼x 2 x2 = 0
1 ∼x 3 x3 = 0
2 ∼x 3
s23(x) s32(x) ⇔ 10 − x1 − x2 10 − x1 − x3 ⇔ x3 x2
x3 = v({3}) = 0 x2 0
s23(x) s32(x) ⇔ 10 − x1 − x2 10 − x1 − x3 ⇔ x3 x2
x2 = v({2}) = 0 x3 0
s23(x) = s32(x) ⇔ 10 − x1 − x2 = 10 − x1 − x3 ⇔ x3 = x2

ͱͳΔͨΊʹ͸

·ͨ͸

ͳΒ͹

·ͨ͸

ͳΒ͹

i ∼x j
sij(x) = sji(x)
sij(x) sji(x) xj = v({j})
sij(x) sji(x) xi = v({i})

14. ਓήʔϜʹ͓͚ΔΧʔωϧͷྫ

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
֤ϓϨΠϠʔͷ࠷େෆຬ͸






x = (x1, x2, x3) ∈
𝒜
(v)
s12(x) = 10 − x1 − x3
s13(x) = 10 − x1 − x2
s21(x) = − x2
s23(x) = 10 − x1 − x2
s31(x) = − x3
s32(x) = 10 − x1 − x3

ͱͳΔͨΊʹ͸ɾɾɾ


ͱͳΔͨΊʹ͸ɾɾɾ


ͱͳΔͨΊʹ͸ɾɾɾ


͸഑෼͔ͩΒ

ΑΓ

Ҏ্ΑΓΧʔωϧ

1 ∼x 2 x2 = 0
1 ∼x 3 x3 = 0
2 ∼x 3 x3 = x2
x = (x1, x2, x3) ∈
𝒜
(v) x1 + x2 + x3 = 10 x1 = 10
𝒦
(v) = {x ∈
𝒜
(v)|(x1, x2, x3) = (10,0,0)}

ͱͳΔͨΊʹ͸

·ͨ͸

ͳΒ͹

·ͨ͸

ͳΒ͹

i ∼x j
sij(x) = sji(x)
sij(x) sji(x) xj = v({j})
sij(x) sji(x) xi = v({i})

15. ήʔϜཧ࿦#4*$ୈճ
Χʔωϧ
࣍ճɿଓɾΧʔωϧ