In this paper we further investigate the well-studied problem of finding a perfect matching in a regular bipartite graph. The first non-trivial algorithm, with running time $O(mn)$, dates back to König's work in 1916 (here $m=nd$ is the number of edges in the graph, $2n$ is the number of vertices, and $d$ is the degree of each node). The currently most efficient algorithm takes time $O(m)$, and is due to Cole, Ost, and Schirra. We improve this running time to $O(\min\{m, \frac{n^{2.5}\ln n}{d}\})$; this minimum can never be larger than $O(n^{1.75}\sqrt{\ln n})$. We obtain this improvement by proving a uniform sampling theorem: if we sample each edge in a $d$-regular bipartite graph independently with a probability $p = O(\frac{n\ln n}{d^2})$ then the resulting graph has a perfect matching with high probability. The proof involves a decomposition of the graph into pieces which are guaranteed to have many perfect matchings but do not have any small cuts. We then establish a correspondence between potential witnesses to non-existence of a matching (after sampling) in any piece and cuts of comparable size in that same piece. Karger's sampling theorem for preserving cuts in a graph can now be adapted to prove our uniform sampling theorem for preserving perfect matchings. Using the $O(m\sqrt{n})$ algorithm (due to Hopcroft and Karp) for finding maximum matchings in bipartite graphs on the sampled graph then yields the stated running time. We also provide an infinite family of instances to show that our uniform sampling result is tight up to poly-logarithmic factors (in fact, up to $\ln^2 n$).
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。
