Lecture notes for “Analysis of Algorithms”: Maximum matching in general graphs

We present Edmonds’ blossom shrinking algorithm for finding a maximum cardinality matching in a general graph. En route, we obtain an efficient algorithm for finding a minimum vertex cover in a bipartite graph and show that its size is equal to the size of the maximum matching in the graph. We also show that the size of a maximum matching in a general graph is equal to the size of a minimum odd cover of the graph. 1 The maximum matching problem Let G = (V, E) be an undirected graph. A set M ⊆ E is a matching if no two edges in M touch each other or, in other words, if the degree of every vertex in the subgraph (V,M) is at most 1. A vertex v is matched by M if there is an edge of M that touches v. Otherwise, v in unmatched. In the maximum matching problem we are asked to find a matching M in the graph G = (V, E) of maximum size. As we have seen, the maximum matching problem in bipartite graphs can be easily reduced to a maximum flow problem which could then be solved in O(m √ n) time. The maximum matching problem in general, not necessarily bipartite, graphs is more challenging. We present here a classical algorithm of Edmonds [Edm65] for solving the problem. 2 Alternating and augmenting paths If A and B are sets, we let A⊕B = (A−B)∪ (B−A) be their symmetric difference. The following lemma is obvious. ∗School of Computer Science, Tel Aviv University, Tel Aviv 69978, Israel. E–mail: [email protected]