The document describes a maximum flow problem on a network (N,v). It defines the network as N={1,2,3} with capacity values v for various subsets of N. It then defines the maximum flow problem as finding values x1, x2, x3 that satisfy the capacity constraints and maximize the total flow. It finds the maximum flow of 1 by setting x1=0, x2=0, x3=1.

1. ήʔϜཧ࿦#4*$ԋश
ࠨӈͷ‫ۺ‬ήʔϜ

2. ༏Ճ๏ੑΛຬͨ͢ήʔϜʹ͓͚ΔίΞͷఆٛ
ࠨӈͷ‫ۺ‬ήʔϜ

3. ༏Ճ๏ੑΛຬͨ͢ήʔϜʹ͓͚ΔίΞ
ఆཧ
ಛੑؔ਺‫ܗ‬ήʔϜ ʹ͓͍ͯ
͕༏Ճ๏ੑΛຬͨ͢ͱ͖
ίΞ͸
ఆٛɿఏ‫ܞ‬ ͷෆຬ
ಛੑؔ਺‫ܗ‬ήʔϜ ʹ͓͍ͯ
഑෼ ͱఏ‫ܞ‬ ΛͱΔͱ͖
Λ഑෼ ʹର͢Δఏ‫ܞ‬ ͷෆຬͱ͍͏
ίΞʹ‫·ؚ‬ΕΔ഑෼͸ ͱͳΔͨΊ
͢΂ͯͷఏ‫ܞ‬ ʹ͍ͭͯෆຬ͕ͳ͍ঢ়ଶͱ΋‫͑ݴ‬Δ
(N, v) v
C(v) = {x ∈
𝒜
(v)|
∑
i∈S
xi ≥ v(S), ∀S ⊂ N, S ≠ ∅, N}
S
(N, v) x ∈
𝒜
(v) S ⊆ N
e(S, x) = v(S) −
∑
i∈S
xi x S
e(S, x) ≤ 0
S ⊆ N
ఏ‫߹ܞ‬ཧੑ

4. ࠨӈͷ‫͕ۺ‬ἧͬͯॳΊͯՁ஋Λ࣋ͭήʔϜΛߟ͑Δ‫͕ۺ‬ἧͬͨͱ͖ʹ͸
ສԁͷརಘΛಘΔͱ͠
͜Ε͸ৡ౉ՄೳͰ͋Δ
ҎԼͷϓϨΠϠʔ͸ಉ࣭ͷ‫ۺ‬Λ΋ͭͱ͢Δ
ࠨ‫ۺ‬Λͭͣͭ΋ͭϓϨΠϠʔ͕ ਓ
ӈ‫ۺ‬Λͭ΋ͭϓϨΠϠʔ͕ ਓͷ৔߹
ఏ‫ܗܞ‬੒Λߟ͑ͨͱ͖ͷಛੑؔ਺‫ܗ‬ήʔϜͱͯ͠ද‫
͠ݱ‬ίΞΛ‫ٻ‬ΊΑ
ࠨ‫ۺ‬Λͭͣͭ΋ͭϓϨΠϠʔ͕ ਓ
ӈ‫ۺ‬Λͭͣͭ΋ͭϓϨΠϠʔ͕ ਓͷ৔߹
ఏ‫ܗܞ‬੒Λߟ͑ͨͱ͖ͷಛੑؔ਺‫ܗ‬ήʔϜͱͯ͠ද‫
͠ݱ‬ίΞΛ‫ٻ‬ΊΑ
ࠨ‫ۺ‬Λͭͣͭ΋ͭϓϨΠϠʔ͕ ਓ
ӈ‫ۺ‬Λͭ΋ͭϓϨΠϠʔ͕ ਓͷ৔߹
ఏ‫ܗܞ‬੒Λߟ͑ͨͱ͖ͷಛੑؔ਺‫ܗ‬ήʔϜͱͯ͠ද‫
͠ݱ‬ίΞΛ‫ٻ‬ΊΑ
1
2 1
2 2
2 1
ࠨӈͷ‫ۺ‬ήʔϜ
ສԁ
1

5. ࠨ‫ۺ‬Λͭͣͭ΋ͭϓϨΠϠʔ͕ ਓ
ӈ‫ۺ‬Λͭ΋ͭϓϨΠϠʔ͕ ਓͷ৔߹
ఏ‫ܗܞ‬੒Λߟ͑ͨͱ͖ͷಛੑؔ਺‫ܗ‬ήʔϜͱͯ͠ද‫
͠ݱ‬ίΞΛ‫ٻ‬ΊΑ
ϓϨΠϠʔͷू߹Λ ͱ͢ΔϓϨΠϠʔ ͕ࠨ‫ۺ‬Λ΋ͪ
ϓϨΠϠʔ ͕ӈ‫ۺ‬Λ΋ͭͱ͢Δ
ಛੑؔ਺͸ҎԼͷΑ͏ʹͳΔ
2 1
N = {1,2,3} 1,2 3
v({1,2,3}) = 1 v({1,2}) = 0 v({1,3}) = 1 v({2,3}) = 1 v({1}) = 0 v({2}) = 0 v({3}) = 0
ࠨӈͷ‫ۺ‬ήʔϜ
ສԁ
1
ϖΞ ϖΞ ϖΞ

6. ࠨ‫ۺ‬Λͭͣͭ΋ͭϓϨΠϠʔ͕ ਓ
ӈ‫ۺ‬Λͭ΋ͭϓϨΠϠʔ͕ ਓͷ৔߹
ఏ‫ܗܞ‬੒Λߟ͑ͨͱ͖ͷಛੑؔ਺‫ܗ‬ήʔϜͱͯ͠ද‫
͠ݱ‬ίΞΛ‫ٻ‬ΊΑ
ϓϨΠϠʔͷू߹Λ ͱ͢ΔϓϨΠϠʔ ͕ࠨ‫ۺ‬Λ΋ͪ
ϓϨΠϠʔ ͕ӈ‫ۺ‬Λ΋ͭͱ͢Δ
ಛੑؔ਺͸ҎԼͷΑ͏ʹͳΔ
഑෼Λ ͱ͢Δͱ
શମ߹ཧੑ
‫ݸ‬ਓ߹ཧੑ
͓Αͼఏ‫߹ܞ‬ཧੑ͔Β
2 1
N = {1,2,3} 1,2 3
v({1,2,3}) = 1 v({1,2}) = 0 v({1,3}) = 1 v({2,3}) = 1 v({1}) = 0 v({2}) = 0 v({3}) = 0
(x1, x2, y3)
x1 + x2 + y3 = v({1,2,3}) = 1
x1 + x2 ≥ v({1,2}) = 0
x1 + y3 ≥ v({1,3}) = 1
x2 + y3 ≥ v({2,3}) = 1
x1 ≥ v({1}) = 0 x2 ≥ v({2}) = 0 y3 ≥ v({3}) = 0
ࠨӈͷ‫ۺ‬ήʔϜ
ສԁ
1

7. ࠨ‫ۺ‬Λͭͣͭ΋ͭϓϨΠϠʔ͕ ਓ
ӈ‫ۺ‬Λͭ΋ͭϓϨΠϠʔ͕ ਓͷ৔߹
ఏ‫ܗܞ‬੒Λߟ͑ͨͱ͖ͷಛੑؔ਺‫ܗ‬ήʔϜͱͯ͠ද‫
͠ݱ‬ίΞΛ‫ٻ‬ΊΑ
ϓϨΠϠʔͷू߹Λ ͱ͢ΔϓϨΠϠʔ ͕ࠨ‫ۺ‬Λ΋ͪ
ϓϨΠϠʔ ͕ӈ‫ۺ‬Λ΋ͭͱ͢Δ
ಛੑؔ਺͸ҎԼͷΑ͏ʹͳΔ
഑෼Λ ͱ͢Δͱ
શମ߹ཧੑ
‫ݸ‬ਓ߹ཧੑ
͓Αͼఏ‫߹ܞ‬ཧੑ͔Β
Ώ͑ʹίΞ͸ ͷ഑෼ͷΈ‫ؚ‬Ή
2 1
N = {1,2,3} 1,2 3
v({1,2,3}) = 1 v({1,2}) = 0 v({1,3}) = 1 v({2,3}) = 1 v({1}) = 0 v({2}) = 0 v({3}) = 0
(x1, x2, y3)
x1 + x2 + y3 = v({1,2,3}) = 1
x1 + x2 ≥ v({1,2}) = 0
x1 + y3 ≥ v({1,3}) = 1
x2 + y3 ≥ v({2,3}) = 1
x1 ≥ v({1}) = 0 x2 ≥ v({2}) = 0 y3 ≥ v({3}) = 0
(0,0,1)
ࠨӈͷ‫ۺ‬ήʔϜ
ສԁ
1
1 − y3 ≥ 0 ⇔ 1 ≥ y3
1 − x2 ≥ 1 ⇔ 0 ≥ x2
1 − x1 ≥ 1 ⇔ 0 ≥ x1
x1 = 0
x2 = 0
y3 = 1

8. ࠨ‫ۺ‬Λͭͣͭ΋ͭϓϨΠϠʔ͕ ਓ
ӈ‫ۺ‬Λͭͣͭ΋ͭϓϨΠϠʔ͕ ਓͷ৔߹
ఏ‫ܗܞ‬੒Λߟ͑ͨͱ͖ͷಛੑؔ਺‫ܗ‬ήʔϜͱͯ͠ද‫
͠ݱ‬ίΞΛ‫ٻ‬ΊΑ
ϓϨΠϠʔͷू߹Λ ͱ͢ΔϓϨΠϠʔ ͕ࠨ‫ۺ‬Λ΋ͪ
ϓϨΠϠʔ ͕ӈ‫ۺ‬Λ΋ͭͱ͢Δ
ಛੑؔ਺͸ҎԼͷΑ͏ʹͳΔ
2 2
N = {1,2,3,4} 1,2 3,4
v({1,2,3,4}) = 2 v({1,2,3}) = 1 v({1,2,4}) = 1 v({1,3,4}) = 1 v({2,3,4}) = 1
v({1,2}) = 0 v({1,3}) = 1 v({1,4}) = 1 v({2,3}) = 1 v({2,4}) = 1 v({3,4}) = 0
v({1}) = 0 v({2}) = 0 v({3}) = 0 v({4}) = 0
ࠨӈͷ‫ۺ‬ήʔϜ

9. ϓϨΠϠʔͷू߹Λ ͱ͢ΔϓϨΠϠʔ ͕ࠨ‫ۺ‬Λ΋ͪ
ϓϨΠϠʔ ͕ӈ‫ۺ‬Λ΋ͭͱ͢Δ
ಛੑؔ਺͸ҎԼͷΑ͏ʹͳΔ
഑෼Λ ͱ͢Δͱ
શମ߹ཧੑ
‫ݸ‬ਓ߹ཧੑ
͓Αͼఏ‫߹ܞ‬ཧੑ͔Β
N = {1,2,3,4} 1,2 3,4
v({1,2,3,4}) = 2 v({1,2,3}) = 1 v({1,2,4}) = 1 v({1,3,4}) = 1 v({2,3,4}) = 1
v({1,2}) = 0 v({1,3}) = 1 v({1,4}) = 1 v({2,3}) = 1 v({2,4}) = 1 v({3,4}) = 0
v({1}) = 0 v({2}) = 0 v({3}) = 0 v({4}) = 0
(x1, x2, y3, y4)
x1 + x2 + y3 + y4 = 2
x1 + x2 + y3 ≥ 1 x1 + x2 + y4 ≥ 1 x1 + y3 + y4 ≥ 1 x2 + y3 + y4 ≥ 1
x1 + x2 ≥ 0 x1 + y3 ≥ 1 x1 + y4 ≥ 1 x2 + y3 ≥ 1 x2 + y4 ≥ 1 y3 + y4 ≥ 0
x1 ≥ 0 x2 ≥ 0 y3 ≥ 0 y4 ≥ 0
ࠨӈͷ‫ۺ‬ήʔϜ

10. ϓϨΠϠʔͷू߹Λ ͱ͢ΔϓϨΠϠʔ ͕ࠨ‫ۺ‬Λ΋ͪ
ϓϨΠϠʔ ͕ӈ‫ۺ‬Λ΋ͭͱ͢Δ
ಛੑؔ਺͸ҎԼͷΑ͏ʹͳΔ
഑෼Λ ͱ͢Δͱ
શମ߹ཧੑ
‫ݸ‬ਓ߹ཧੑ
͓Αͼఏ‫߹ܞ‬ཧੑ͔Β
Ώ͑ʹ
ίΞ
N = {1,2,3,4} 1,2 3,4
v({1,2,3,4}) = 2 v({1,2,3}) = 1 v({1,2,4}) = 1 v({1,3,4}) = 1 v({2,3,4}) = 1
v({1,2}) = 0 v({1,3}) = 1 v({1,4}) = 1 v({2,3}) = 1 v({2,4}) = 1 v({3,4}) = 0
v({1}) = 0 v({2}) = 0 v({3}) = 0 v({4}) = 0
(x1, x2, y3, y4)
x1 + x2 + y3 + y4 = 2
x1 + x2 + y3 ≥ 1 x1 + x2 + y4 ≥ 1 x1 + y3 + y4 ≥ 1 x2 + y3 + y4 ≥ 1
x1 + x2 ≥ 0 x1 + y3 ≥ 1 x1 + y4 ≥ 1 x2 + y3 ≥ 1 x2 + y4 ≥ 1 y3 + y4 ≥ 0
x1 ≥ 0 x2 ≥ 0 y3 ≥ 0 y4 ≥ 0
C(v) = {x ∈
𝒜
(v)|x1 = x2 = X, y3 = y4 = Y, X + Y = 1}
ࠨӈͷ‫ۺ‬ήʔϜ
0 ≤ x1 ≤ 1 0 ≤ x2 ≤ 1
0 ≤ y3 ≤ 1 0 ≤ y4 ≤ 1
x1 + y3 = 1 x2 + y4 = 1
x1 + y4 = 1 x2 + y3 = 1
x1 = x2 = X y3 = y4 = Y
͔ͭ ΑΓ
x1 + y3 = 1 x2 + y3 = 1
x1 + x2 + 2y3 = 2 = x1 + x2 + y3 + y4 ⇔ y3 = y4

11. ࠨ‫ۺ‬Λͭͣͭ΋ͭϓϨΠϠʔ͕ ਓ
ӈ‫ۺ‬Λͭ΋ͭϓϨΠϠʔ͕ ਓͷ৔߹
ఏ‫ܗܞ‬੒Λߟ͑ͨͱ͖ͷಛੑؔ਺‫ܗ‬ήʔϜͱͯ͠ද‫
͠ݱ‬ίΞΛ‫ٻ‬ΊΑ
ϓϨΠϠʔͷू߹Λ ͱ͢ΔϓϨΠϠʔ ͕ࠨ‫ۺ‬Λ΋ͪ
ϓϨΠϠʔ ͕ӈ‫ۺ‬Λ̎ͭ΋ͭͱ͢Δ
ಛੑؔ਺͸ҎԼͷΑ͏ʹͳΔ
2 1
N = {1,2,3} 1,2 3
v({1,2,3}) = 2 v({1,2}) = 0 v({1,3}) = 1 v({2,3}) = 1 v({1}) = 0 v({2}) = 0 v({3}) = 0
ࠨӈͷ‫ۺ‬ήʔϜ

12. ࠨ‫ۺ‬Λͭͣͭ΋ͭϓϨΠϠʔ͕ ਓ
ӈ‫ۺ‬Λͭ΋ͭϓϨΠϠʔ͕ ਓͷ৔߹
ఏ‫ܗܞ‬੒Λߟ͑ͨͱ͖ͷಛੑؔ਺‫ܗ‬ήʔϜͱͯ͠ද‫
͠ݱ‬ίΞΛ‫ٻ‬ΊΑ
ϓϨΠϠʔͷू߹Λ ͱ͢ΔϓϨΠϠʔ ͕ࠨ‫ۺ‬Λ΋ͪ
ϓϨΠϠʔ ͕ӈ‫ۺ‬Λ̎ͭ΋ͭͱ͢Δ
ಛੑؔ਺͸ҎԼͷΑ͏ʹͳΔ
2 1
N = {1,2,3} 1,2 3
v({1,2,3}) = 2 v({1,2}) = 0 v({1,3}) = 1 v({2,3}) = 1 v({1}) = 0 v({2}) = 0 v({3}) = 0
ࠨӈͷ‫ۺ‬ήʔϜ
ϖΞ ϖΞ ϖΞ

13. ࠨ‫ۺ‬Λͭͣͭ΋ͭϓϨΠϠʔ͕ ਓ
ӈ‫ۺ‬Λͭ΋ͭϓϨΠϠʔ͕ ਓͷ৔߹
ఏ‫ܗܞ‬੒Λߟ͑ͨͱ͖ͷಛੑؔ਺‫ܗ‬ήʔϜͱͯ͠ද‫
͠ݱ‬ίΞΛ‫ٻ‬ΊΑ
ϓϨΠϠʔͷू߹Λ ͱ͢ΔϓϨΠϠʔ ͕ࠨ‫ۺ‬Λ΋ͪ
ϓϨΠϠʔ ͕ӈ‫ۺ‬Λ̎ͭ΋ͭͱ͢Δ
ಛੑؔ਺͸ҎԼͷΑ͏ʹͳΔ
഑෼Λ ͱ͢Δͱ
શମ߹ཧੑ
‫ݸ‬ਓ߹ཧੑ
͓Αͼఏ‫߹ܞ‬ཧੑ͔Β
2 1
N = {1,2,3} 1,2 3
v({1,2,3}) = 2 v({1,2}) = 0 v({1,3}) = 1 v({2,3}) = 1 v({1}) = 0 v({2}) = 0 v({3}) = 0
(x1, x2, y3)
x1 + x2 + y3 = v({1,2,3}) = 2
x1 + x2 ≥ v({1,2}) = 0
x1 + y3 ≥ v({1,3}) = 1
x2 + y3 ≥ v({2,3}) = 1
x1 ≥ v({1}) = 0 x2 ≥ v({2}) = 0 y3 ≥ v({3}) = 0
ࠨӈͷ‫ۺ‬ήʔϜ

14. ࠨ‫ۺ‬Λͭͣͭ΋ͭϓϨΠϠʔ͕ ਓ
ӈ‫ۺ‬Λͭ΋ͭϓϨΠϠʔ͕ ਓͷ৔߹
ఏ‫ܗܞ‬੒Λߟ͑ͨͱ͖ͷಛੑؔ਺‫ܗ‬ήʔϜͱͯ͠ද‫
͠ݱ‬ίΞΛ‫ٻ‬ΊΑ
ϓϨΠϠʔͷू߹Λ ͱ͢ΔϓϨΠϠʔ ͕ࠨ‫ۺ‬Λ΋ͪ
ϓϨΠϠʔ ͕ӈ‫ۺ‬Λ̎ͭ΋ͭͱ͢Δ
ಛੑؔ਺͸ҎԼͷΑ͏ʹͳΔ
഑෼Λ ͱ͢Δͱ
શମ߹ཧੑ
‫ݸ‬ਓ߹ཧੑ
͓Αͼఏ‫߹ܞ‬ཧੑ͔Β
Ώ͑ʹ
ίΞ
2 1
N = {1,2,3} 1,2 3
v({1,2,3}) = 2 v({1,2}) = 0 v({1,3}) = 1 v({2,3}) = 1 v({1}) = 0 v({2}) = 0 v({3}) = 0
(x1, x2, y3)
x1 + x2 + y3 = v({1,2,3}) = 2
x1 + x2 ≥ v({1,2}) = 0
x1 + y3 ≥ v({1,3}) = 1
x2 + y3 ≥ v({2,3}) = 1
x1 ≥ v({1}) = 0 x2 ≥ v({2}) = 0 y3 ≥ v({3}) = 0
C(v) = {x ∈
𝒜
(v)|1 ≤ x1 + y3 ≤ 2, 1 ≤ x2 + y3 ≤ 2,( 0 ≤ x1 ≤ 1, 0 ≤ x2 ≤ 1,) 0 ≤ y3 ≤ 2}
ࠨӈͷ‫ۺ‬ήʔϜ
2 − y3 ≥ 0 ⇔ 2 ≥ y3
2 − x2 ≥ 1 ⇔ 1 ≥ x2
2 − x1 ≥ 1 ⇔ 1 ≥ x1
0 ≤ y3 ≤ 2
0 ≤ x2 ≤ 1
0 ≤ x1 ≤ 1

15. ࠨ‫ۺ‬Λͭͣͭ΋ͭϓϨΠϠʔ͕ ਓ
ӈ‫ۺ‬Λͭ΋ͭϓϨΠϠʔ͕ ਓͷ৔߹
ఏ‫ܗܞ‬੒Λߟ͑ͨͱ͖ͷಛੑؔ਺‫ܗ‬ήʔϜͱͯ͠ද‫
͠ݱ‬ίΞΛ‫ٻ‬ΊΑ
ίΞ͸ ͷ഑෼ͷΈ‫ؚ‬Ή
ࠨ‫ۺ‬Λͭͣͭ΋ͭϓϨΠϠʔ͕ ਓ
ӈ‫ۺ‬Λͭ΋ͭϓϨΠϠʔ͕ ਓͷ৔߹
ఏ‫ܗܞ‬੒Λߟ͑ͨͱ͖ͷಛੑؔ਺‫ܗ‬ήʔϜͱͯ͠ද‫
͠ݱ‬ίΞΛ‫ٻ‬ΊΑ
2 1
(0,0,1)
2 1
C(v) = {x ∈
𝒜
(v)|1 ≤ x1 + y3 ≤ 2, 1 ≤ x2 + y3 ≤ 2,( 0 ≤ x1 ≤ 1, 0 ≤ x2 ≤ 1,) 0 ≤ y3 ≤ 2}
ࠨӈͷ‫ۺ‬ήʔϜ
1
2 3
x1 ≤ 0
x2 ≤ 0
x3 ≤ 1
ίΞ
1
2 3
x1 ≤ 1
x2 ≤ 1
y3 ≤ 2
ίΞ
ϓϨΠϠʔ͸ަবͰܾ·ΔӈͷήʔϜΑΓ
ӈ‫ۺ‬Λͭഁ‫͢غ‬Δ͜ͱͰ
ࠨͷήʔϜʹ࣮ͯ֬͠ʹΛಘΔ͜ͱ΋Մೳ

16. ήʔϜཧ࿦#4*$ԋश
ࠨӈͷ‫ۺ‬ήʔϜ
࣍ճɿԋश