Maximum matchings in sparse random graphs: Karp-Sipser revisited

We study the average performance of a simple greedy algorithm for finding a matching in a sparse random graph G(n,c/n), where c > 0 is constant. The algorithm was first proposed by Karp and Sipser [Proceedings of the Twenty-Second Annual IEEE Symposium on Foundations of Computing, 1981, pp. 364-375]. We give significantly improved estimates of the errors made by the algorithm. For the subcritical case where c < e we show that the algorithm finds a maximum matching with high probability. If c > e then with high probability the algorithm produces a matching which is within n(1/5+o(1)) of maximum size. (C) 1998 John Wiley & Sons, Inc.