Bipartite Perfect Matching is in quasi-NC

TL;DR

This paper demonstrates that the bipartite perfect matching problem can be solved in quasi-NC$^2$, showing it has uniform circuits of quasi-polynomial size and poly-logarithmic depth, advancing parallel complexity understanding.

Contribution

It establishes the bipartite perfect matching problem is in quasi-NC$^2$, improving from previous exponential circuit size bounds by nearly derandomizing the Isolation Lemma.

Findings

Bipartite perfect matching is in quasi-NC$^2$.

Circuit size for bipartite perfect matching is quasi-polynomial.

Derandomization of the Isolation Lemma enables this result.

Abstract

We show that the bipartite perfect matching problem is in quasi-NC$^2$. That is, it has uniform circuits of quasi-polynomial size $n^{O(\log n)}$, and $O(log^2 n)$ 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.