Bipartite Perfect Matching is in quasi-NC

Stephen Fenner, Rohit Gurjar, Thomas Thierauf

We show that the bipartite perfect matching problem is in QuasiNC2. That is, it has uniform circuits of quasi-polynomial size nO(log n), and O(log2 n) depth. Previously, only an exponential upper bound was known on the size of such circuits with poly-logarithmic depth.

We obtain our result by an almost complete derandomization of the famous Isolation Lemma when applied to yield an efficient randomized parallel algorithm for the bipartite perfect matching problem.