Parallel approximate matching

Abstract

The author solves the maximum cardinality matching problem approximately. Determining the parallel complexity of maximum cardinality matching in bipartite graphs is a famous open problem in parallel algorithm design. The problem is known to be in RNC, but all known fast parallel algorithms that find maximum cardinality matchings require the use of random numbers. They are based on matrix algebra, and are inherently efficient for sparse graphs. Therefore, the problem of finding an approximate maximum cardinality matching is considered. The parallel matching algorithm of A. Goldberg et al. (1988) can be modified so that it runs in O(a/sup 2/ log/sup 3/ n) time on an exclusive read exclusive write (EREW) parallel random access machine (PRAM) with n+m processors and finds a matching of size (1-1/a)p when given a graph with n vertices, m edges and a maximum cardinality matching of size p. The resulting algorithm is deterministic. >