ABSTRACT:We study the average performance of a simple greedy algorithm for finding a matching in a sparse random graph G , where c ) 0 is constant. The algorithm was first n, c r n w proposed by Karp and Sipser Proceedings of the Twenty-Second Annual IEEE Symposium on x 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 1r5qoŽ1. of maximum size.

We consider the following randomized algorithm for finding a matching M in an arbitrary graph G = (V, E ) . Repeatedly, choose a random vertex u , then a random neighbour u of u . Add edge { u , u ) to M and delete vertices u, u from G along with any vertices that become isolated. Our main result is that there exists a positive constant E such that the expected ratio of the size of the matching produced to the size of largest matching in G is at least 0.5 + e. We obtain stronger results for sparse graphs and trees and consider extensions to hypergraphs. 0 1995 John Wiley & Sons, Inc.

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