割当問題とマッチング

Assignment Problem B=(b ij ) : n×n matrix C ⊆ {b ij |1 ≦ i ≦ n, 1 ≦ j ≦ n} C is diagonal ⇔ (1) ∀ b ij, b kh ∈ C, ⇒ i≠k and j≠h (2) |C | = n def max (min) {Σb ij | C is diagonal} C

max-weight bipartite matching problem Given G=(S ∪ T, A) complete bipartite graph, |S|=|T|=ncomplete bipartite graph, w:A→R arc weight find min{Σ w(a) | K : perfect matching}perfect matching a∈Ka∈K max-weight bipartite matching problem assignment problem EQUIVALENT

Bipartite graph G=(V, A) is bipartite ⇔ ∃ partition of V, V=S ∪ T s.t. ∃ arc which has both their vertices in S, ∃ arc which has both their vertices in T def ・ G=(S ∪ T, A) is complete bipartite ⇔ A=S×T def

Matching G=(V, A) M ⊆ A M is a matching ⇔ ∀ a, b ∈ M a and b do not have any vertex in common. M is a perfect matching ⇔ any vertex in V is covered by an arc in M def

diagonal ⇔ perfect matching Assume that any arc weigh ( matrix element) is given by nonnegative real number. S T

Node weighting (potential) G=(S ∪ T, A) u(i) i ∈ S v(j) j ∈ T u, v is feasible node weighting ⇔ u(i)+v(j) ≧ w(i, j) ∀ {i, j} ∈ A def ex. u(i) = 0, ∀ i ∈ S v(j)=max{w(i, j) |i ∈ S}, ∀ j ∈ T is a feasible node weighting u\v

Lemma 6.1 u, v: feasible node weighting K : perfect matching = ≦ Corollary 6.2 u, v: feasible node weighting A = ={{i, j} ∈ A | u(i)+v(j)=w(i, j)} K ⊆ A = K : perfect matching ⇒ K : max-weight perfect matching

Reduced weighting w(i, j) = u(i)+v(j)-w(i, j) ∀ {i, j} ∈ A u, v is feasible node weighting ⇔ w(i, j) ≧ 0 ∀ {i, j} ∈ A

Graph orientation u, v: feasible node weighting K : (any) matching S T orientation i ∈ S, j ∈ T {i, j} ∈ K ⇒ j→i {i, j} ∈ A\K ⇒ i→j add new vertices s, t, and new arcs {(s, i) | i ∈ S\K} ∪ {(j, t) |j ∈ T\K} s t Each vertex i ∈ S has one incoming arcs

Re-weighting u, v: feasible node weighting K : (any) matching S T re-weighting {i, j} ∈ K ⇒ l(j, i) = -w(i, j) {i, j} ∈ A\K ⇒ l(i. j) = w(i, j) l(s, i) = 0 ∀ i ∈ S\K l(j, t)=0 ∀ j ∈ T\K s t d(v) : distance from s to v with respect to l Property u’(i) = u(i)+d(i), i ∈ S v’(j) = v(j) - d(j), j ∈ T : feasible node potential

feasible node weighting? re-weighting {i, j} ∈ K ⇒ l(j, i) = -w(i, j) {i, j} ∈ A\K ⇒ l(i. j) = w(i, j) l(s, i) = 0 ∀ i ∈ S\K l(j, t)=0 ∀ j ∈ T\K u’(i) = u(i)+d(i), i ∈ S v’(j) = v(j) - d(j), j ∈ T feasible? ( u’(i)+v’(j) ≧ w(i, j) ) {i, j} ∈ A\K i j w(i, j) d(j) ≦ d(i)+l(i, j) ＝ d(i)+w(i, j) ∴ u’(i)+v’(j) ＝ u(i)+d(i)+v(j)-d(j) ≧ u(i)+v(j)-w(i, j) ＝ u(i)+v(j)-(u(i)+v(j)-w(i, j)) ＝ w(i, j) {i, j} ∈ K i j -w(i, j) d(i)=d(j)+l(i, j) =d(j)-w(i, j) ∵ i has exactly one incoming arc ∴ u’(i)+v’(j) ＝ u(i)+v(j)-w(i, j) ＝ w(i, j)

Corollary P : shortest path from s to t with respect to l {i, j} ∈ A(P) ⇒ u’(i)+v’(j)=w(i, j) K △ A(P) = (K\A(P)) ∪ (A(P)\K) : matching s t s t | K △ A(P) |>|K|

Algorithm (successive shortest path) Maintain a feasible node weighting u, v and a matching K ⊆ A = ={{i, j} ∈ A | u(i)+v(j)+w(i, j)} 1.Orient each arc and add new vertices and arcs. 2.Find a shortest path P and shortest path distance d (since l ≧ 0, we can use Dijkstra algorithm) 2.Update u and v by d 3.Update K to K △ A(P) Repeat the following steps: Each step increases the cardinality of K. Thus the algorithm terminates at most n iterations.

Example (1) Initialization w u(i) = 0, ∀ i ∈ S,v(j)=max{w(i, j) |i ∈ S}, ∀ j ∈ T arcs in A = K

Example (2) Iteration 1 w s t Find d shortest path P update

Example (3) Iteration 2 w s t Find d shortest path P update optimal

LP variable x :A→{0, 1} 1 ( when an optimal matching contains an arc a) x(a) = 0 ( otherwise)

LP -dual

Hichcock’s Transportation Problem 輸送問題

Problem Definition Given G=(S ∪ T, A) bipartite graph supply b(i) i ∈ S, demand b(j), j ∈ T (Σb(i) = Σb(j) ) arc cost c(i, j), {i, j} ∈ A i j warehouse customer Determine a way of distribution of goods from each warehouse to each customoer with minimizing the total transpotation cost.

f(i, j) : a value distributed a goods from warehouse i to customer j total transportation cost = Σ c(i, j)f(i, j) {i, j} ∈ A Node cost G=(S ∪ T, A) u(i) i ∈ S v(j) j ∈ T u, v is feasible node cost ⇔ u(i)+v(j) ≦ c(i, j) ∀ {i, j} ∈ A def

Lemma 6.3 u, v: feasible node cost f : feasible distribution Corollary 6.4 u, v: feasible node cost A = ={{i, j} ∈ A | u(i)+v(j)=c(i, j)} If G(S ∪ T, A = ) has a feasible solution f, f is optimal.

Think Think Think Develop an algorithm for Hichcock’s transportation problem Describe an LP formulation of Hichcock’s transportation problem Develop an algorithm for Hichcock’s transportation problem with arc capacity

Matching problem on nonbipartite graphs optimal solution fractional optimal solution value = 18 1/ value = 19