Using Interior-Point Methods for Fast Parallel Algorithms for Bipartite Matching and Related Problems

In this paper interior-point methods for linear programming, developed in the context of sequential computation, are used to obtain a parallel algorithm for the bipartite matching problem. This algorithm finds a maximum cardinality matching in a bipartite graph with n nodes and m edges in $O(\sqrt m \log ^3 n)$ time on a CRCW PRAM. The results here extend to the weighted bipartite matching problem and to the zero-one minimum-cost flow problem, yielding $O(\sqrt m \log ^2 n\log nC)$ algorithms, where $C > 1$ is an upper bound on the absolute value of the integral weights or costs in the two problems, respectively. The results here improve previous bounds on these problems and introduce interior-point methods to the context of parallel algorithm design.