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.
Changes to Revision#1: proof of Lemma 3.4 modified.
Changes to the version#0: Simpler construction of the Isolating weight assignment where the weight size is now improved from $O(\log^3 n)$ bits to $O(\log^2n)$ bits.
Added an RNC algorithm for finding a perfect matching in bipartite graphs with only $O(\log^2n)$ random bits.
We show that the bipartite perfect matching problem is in quasi-NC$^2$. That is, it has uniform circuits of quasi-polynomial size 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.
Simpler construction of the Isolating weight assignment where the weight size is now improved from $O(\log^3 n)$ bits to $O(\log^2 n)$ bits.
Added an RNC algorithm for finding a perfect matching in bipartite graphs with only $O(\log^2 n)$ random bits.
We show that the bipartite perfect matching problem is in quasi-NC$^2$. That is, it has uniform circuits of quasi-polynomial size 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.
