Almost-optimum speed-ups of algorithms for bipartite matching and related problems

We present algorithms for matching and related problems that run on an EREW PRAM with <italic>p</italic> processors. Given is a bipartite graph <italic>G</italic> with <italic>n</italic> vertices, <italic>m</italic> edges, and integral edge costs at most <italic>N</italic> in magnitude. We give an algorithm for the assignment problem (minimum cost perfect bipartite matching) that runs in <italic>&Ogr;</italic>(√<italic>nm</italic> log (<italic>nN</italic>)(log(2<italic>p</italic>))/<italic>p</italic>) time and <italic>&Ogr;</italic>(<italic>m</italic>) space, for <italic>p</italic> ≤ <italic>m</italic>/(√<italic>n</italic>log<supscrpt>2</supscrpt><italic>n</italic>). For <italic>p</italic> = 1 this improves the best known sequential algorithm, and is within a factor of log (<italic>nN</italic>) of the best known bound for the problem without costs (maximum cardinality matching). For <italic>p</italic> > 1 the time is within a factor of log <italic>p</italic> 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.