Parallel Approximate Matching

Determining the parallel complexity of maximum cardinality matching in bipartite graphs is one of the most famous open problems in parallel algorithm design. The problem is known to be in RNC, but all known fast (polylogarithmic running time) parallel algorithms that find maximum cardinality matchings require the use of random numbers. Moreover, they are based on matrix algebra, so they are inherently inefficient for sparse graphs. Therefore, we are interested in the problem of finding an approximate maximum cardinality matching. The parallel matching algorithm of Goldberg, Plotkin and Vaidya (FOCS 1988) can be modified so that it runs in O(a2log3n) time on an EREW 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.