ゲーム理論BASIC 第35回 -交渉集合-
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The document describes an algorithm for solving maximum weight independent set problems on graphs. It defines several notations and concepts used in the algorithm, including the graph (N, v), the set of feasible solutions A(v), and the optimal solution set C(v). It then provides an example graph with nodes 1, 2, 3 and sets constraints on the node weights and optimal solution. The algorithm works by starting from a feasible but non-optimal solution x, and iteratively finding neighboring solutions (y, S) and (z, T) that improve on x according to the problem constraints until an optimal solution is reached.
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ゲーム理論BASIC 第35回 -交渉集合-
- 2. ҟٞ ٯҟٞ ަবू߹ ίΞͱަবू߹ ਓήʔϜʹ͓͚Δަবू߹ͷྫ
- 5. ҟٞ ҟٞ͋Γ ަবΛҟٞɾٯҟٞͱ͍͏ͰܗఆࣜԽ ఆٛɿҟٞ ಛੑؔܗήʔϜ  ʹ͓͍ͯ  ͱਓͷϓϨΠϠʔ  ΛͱΔ ࠓ   ͳΔఏܞ  ͱ  ͕ଘࡏͯ͠   ͷ̎ͭͷ͕݅ຬͨ͞ΕͯΔͱ͢Δ ͜ͷͱ͖  ʹ͓͍ͯϓϨΠϠʔ   ʹରͯ͠ҟٞ  Λͭͱ͍͏ (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
- 7. ٯҟٞ ҟٞ͋Γ ަবΛҟٞɾٯҟٞͱ͍͏ͰܗఆࣜԽ ఆٛɿٯҟٞ  ͷ  ʹର͢Δҟٞ  ʹରͯ͠   ͳΔఏܞ  ͱ  ͕ଘࡏͯ͠    ͷͭͷ͕݅ຬͨ͞ΕͯΔͱ͢Δ ͜ͷͱ͖ϓϨΠϠʔ   ͷҟٞ  ʹରͯ͠ٯҟٞ  Λͭͱ͍͏ 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)
- 9. ίΞͱަবू߹ ఆཧɿίΞͱަবू߹ͷؔ ήʔϜ  ʹ͓͍ͯ ίΞΛ  ަবू߹Λ  ͱ͢Δͱ  ˎަবू߹ৗʹඇۭͰ͋Δ ผػձʹূ໌ ূ໌ ίΞ  Ͱ͋Ε໌Β͔  ͱ͢Δ  Ͱͳ͍ͱԾఆ͠  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 ϓϨΠϠʔͷٯҟٞଘࡏ͢Δ͔ʁ/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)