SIAM Journal on Computing, Ahead of Print.
We show that the bipartite perfect matching problem is in quasi-$\mathsf{NC}^2$. 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.

中文翻译：

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两方完美匹配在准NC中

《 SIAM计算杂志》，预印本。
我们证明二元完全匹配问题在准$ \ mathsf {NC} ^ 2 $中。即，它具有准多项式大小$ n ^ {O（\ log n）} $和深度$ O（\ log ^ 2 n）$的均匀电路。以前，在具有多对数深度的此类电路的大小上，仅知道指数上限。我们通过对著名的隔离引理进行几乎完全的去随机化来获得我们的结果，当将其应用于产生针对二分法完美匹配问题的高效随机并行算法时。