Perfect Bipartite Matching in Pseudo-Deterministic RNC

In this paper we present a pseudo-deterministic RNC algorithm for finding perfect matchings in bipartite graphs. Specifically, our algorithm is a randomized parallel algorithm which uses poly(n) processors, poly(log n) depth, poly(log n) random bits, and outputs for each bipartite input graph a unique perfect matching with high probability. That is, it returns the same matching for almost all random seeds. Our work improves upon different aspects of prior work. The celebrated works of Karp, Uval, Wigderson [13] and Mulmuley, Vazirani, Vazirani [15] which find perfect matchings in RNC produce different matchings on different executions. The recent work of Fenner, Gurjar, and Thierauf [7] shows a deterministic parallel algorithm for bipartite perfect matching but requires 2 (quasi-polynomially many) processors, proving that bipartite matching is in quasi-NC. Our algorithm is the first algorithm to return unique perfect matchings with only polynomially many processors. As an immediate consequence we also find a pseudo-deterministic RNC algorithm for depth first search (DFS).