We consider the well-studied problem of finding a perfect matching in $d$-regular bipartite graphs with $2n$ vertices and $m = nd$ edges. While the best-known algorithm for general bipartite graphs (due to Hopcroft and Karp) takes $O(m \sqrt{n})$ time, in regular bipartite graphs, a perfect matching is known to be computable in $O(m)$ time. Very recently, the $O(m)$ bound was improved to $O(\min\{m, \frac{n^{2.5}\ln n}{d}\})$ expected time, an expression that is bounded by $\tilde{O}(n^{1.75})$. In this paper, we further improve this result by giving an $O(\min\{m, \frac{n^2\ln^3 n}{d}\})$ expected time algorithm for finding a perfect matching in regular bipartite graphs; as a function of $n$ alone, the algorithm takes expected time $O((n\ln n)^{1.5})$.
To obtain this result, we design and analyze a two-stage sampling scheme that reduces the problem of finding a perfect matching in a regular bipartite graph to the same problem on a subsampled bipartite graph with $O(n\ln n)$ edges that has a perfect matching with high probability. The matching is then recovered using the Hopcroft-Karp algorithm. While the standard analysis of Hopcroft-Karp gives us an $\tilde{O}(n^{1.5})$ running time, we present a tighter analysis for our special case that results in the stronger $\tilde{O}(\min\{m, \frac{n^2}{d} \})$ time mentioned earlier.
Our proof of correctness of this sampling scheme uses a new correspondence theorem between cuts and Hall's theorem ``witnesses'' for a perfect matching in a bipartite graph that we prove. We believe this theorem may be of independent interest; as another example application, we show that a perfect matching in the support of an $n \times n$ doubly stochastic matrix with $m$ non-zero entries can be found in expected time $\tilde{O}(m + n^{1.5})$.
Deterministically Isolating a Perfect Matching in Bipartite Planar Graphs
我们提出了一个确定性的方式分配小（日志位）的权重的二分平面图的边缘，使最小重量完美匹配成为唯一的。隔离引理（Mulmuley et al. 1987）中所描述的实现了相同的一般图使用随机加权方案，而我们可以做到这一点时，确定性限制到二分平面图。因此，我们减少了决策和建设版本的匹配问题，以测试是否矩阵是奇异的，根据承诺，其行列式是0或1，从而获得一个高度并行的SPL算法的二分平面图。这改进了先前已知的非均匀SPL的界限（Allender et al. 1999）和$NC^2$的界限（Miller and Naor 1995，Mahajan and Varadarajan 2000）。这也重新点燃了希望，获得一个确定性的并行算法，构造一个完美的匹配在非二部平面图，这已经开放了很长一段时间。我们的技术是基本和简单的。
