Maximum Matching Algorithm

Transcription

1 Maximum Matching Algorithm Thm: M is maximum no M-augmenting path How to find efficiently? 1

2 Edmonds Blossom Algorithm In bipartite graphs, we can search quickly for augmenting paths because we explore from each vertex at most once. An M-alternating path from u can reach a vertex x in the same partite set as u only along a saturated edge. Hence only once can we search and explore x. A B C D Augmenting path: A-F-B-G-C-H The path arrives at B and C along saturated edges FB and GC, respectively E F G H 2

3 Edmonds Blossom Algorithm This property fails in graphs with odd cycles, because M-alternating paths from an unsaturated vertex may reach x both along saturated and along unsaturated edges. 3

4 Stem+ Blossom=Flower A flower with respect to a matching M is composed of a stem, which is an alternating path of even length from an unsaturated vertex r, called root, to a vertex b, and an alternating cycle of odd length that passes through b, called blossom. The last edge on the stem belongs to M. The two edges of the blossom incident on b are not in M. The vertex b is called the base of the blossom. 4

5 Flower, Stem, Blossom That M be a matching in a graph G, and let u be an M-unsaturated vertex. A flower is the union of two M-alternating paths from u that reach a vertex x on steps of opposite parity (having not done so earlier). The stem of the flower is the maximal common initial path (of nonnegative even length). The blossom of the flower is the odd cycle obtained by deleting the stem. 5

6 Example of Flower, Stem, and Blossom Flower Stem Blossom 6

7 Shrinking a blossom Definition (Shrinking a blossom) Given a graph G = (V,E) with a matching M and a blossom B, the shrunk graph G/B with matching M/B is defined as follows: V (G/B) = (V \ B) {b} E(G/B) = E \ E[B] ={xy in E s.t. x and y are not in B} U { by bý in E where b in B and y in V\B} M/B = M \ E[B], where E[B] denotes the set of edges within B, and b is a new vertex disjoint from V. 7

8 Shrinking Contd... Observe that M/B is a matching in G, because the definition of a blossom precludes the possibility that M contains more than one edge with one but not both endpoints in B. Observe also that G/B may contain parallel edges between vertices, if G contains a vertex which is joined to B by more than one edge. The relation between matchings in G and matchings in G/B is summarized by the following Theorem: Let M be a matching of G, and let B be an M-blossom. Then, M is a maximum-size matching if and only if M/B is a maximum-size matching in G/B. 8

9 Finding an M-augmenting path Find an alternating walk between two free vertices. This can be done in linear time by a BFS. Either an M-augmenting path or a blossom can be found. If a blossom is found, shrink it, and (recursively) find an M/C-augmenting path P in G/C, and then expand P to an M-augmenting path in G. 9

10 Example Start point a Blossom c f g Contract Blossom {c, e, f} a C g u e h u h b d b d x x Find a new Blossom {u,a,b,c,d} 10

11 Example Augmenting Path in original graph a c f Contract Blossom {u,a,b,c,d} U g h x u x b d e u x a b C d New Augmenting Path!!! Expand Blossom U U x 11

12 Example Original Matching New Matching f g f g u a c e h u a c e h x b d x b d 12

13 Edmonds Blossom Algorithm Input: A graph G, a matching M in G, an M-unsaturated vertex u. 13

14 Idea: Edmonds Blossom Algorithm Explore M-altenating paths from u, recording for each vertex the vertex from which it was reached Contract blossoms when found Maintain sets S and T, with S consisting of u and the vertices reached along saturated edges. Reach an unsaturated vertex yields an augmentation 14

15 Edmonds Blossom Algorithm Initialization: S={u} and T= 15

16 Edmonds Blossom Algorithm Iteration: step 1 If S has no unmarked vertex, stop There is no M-augmenting path from u Otherwise, select an unmarked v S. To explore from v, successively consider each y N(v) such that y T. 16

17 Edmonds Blossom Algorithm Iteration: step 2 If y is unsaturdated by M, then trace back from y (expanding blossoms as needed) to report an M-augmrnting u,y-path. 17

18 Edmonds Blossom Algorithm Iteration: step 3 If y S, then a blossom has been found. Suspend the exploration of v and contract the blossom, replacing its vertices in S and T by a single new vertex in S. Continue the search from this vertex in the smaller graph. Otherwise, y is matched to some w by M. Include y in T (reached from v), and include w in S (reached from y). 18

19 Edmonds Blossom Algorithm Iteration: step 4 After exploring all such neighbors of v, mark v and iterate. 19