We present algorithms for matching and related problems that run on an EREW PRAM with p processors. Given is a bipartite graph G with n vertices, m edges, and integral edge costs at most N in magnitude. We give an algorithm for the assignment problem (minimum cost perfect bipartite matching) that runs in O(√nm log (nN)(log(2p))/p) time and O(m) space, for p ≤ m/(√nlog2n). For p = 1 this improves the best known sequential algorithm, and is within a factor of log (nN) of the best known bound for the problem without costs (maximum cardinality matching). For p > 1 the time is within a factor of log p of optimum speed-up. Extensions include an algorithm for maximum cardinality bipartite matching with slightly better processor bounds, and similar results for bipartite degree-constrained subgraph problems (with and without costs). Our ideas also extend to general graph matching problems.
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Gabow et al. (1988) studied this question.