arXiv 0902.16172009-02cs.DScs.DM

正则二部图的n（n^{1.5}）时间完美匹配

Perfect Matchings in Õ(n^{1.5}) Time in Regular Bipartite Graphs

Ashish Goel, Michael Kapralov, Sanjeev Khanna

备注：: Added analysis of the Hopcroft-Karp algorithm on the subsampled graph

摘要：

我们考虑了在$d$ -正则二部图中寻找完美匹配的问题，该图具有$2n$顶点和$m = nd$边。一般二部图的最著名算法（由Hopcroft和Karp提出）需要$O(m \sqrt{n})$时间，而在规则二部图中，完美匹配已知可以在$O(m)$时间内计算。最近，$O(m)$界限被改进为$O(\min\{m, \frac{n^{2.5}\ln n}{d}\})$预期时间，这是一个受$\tilde{O}(n^{1.75})$界限的表达式。在本文中，我们进一步改进了这一结果，给出了一个$O(\min\{m, \frac{n^2\ln^3 n}{d}\})$期望时间算法，在正则二部图中找到一个完美的匹配，作为一个功能的$n$单独，该算法需要的期望时间$O((n\ln n)^{1.5})$。为了获得这个结果，我们设计和分析了一个两阶段的抽样方案，减少了在一个经常的二分图中找到一个完美匹配的问题，以同样的问题上的二分图与$O(n\ln n)$边缘，具有高概率的完美匹配。然后使用Hopcroft-Karp算法恢复匹配。虽然Hopcroft-Karp的标准分析为我们提供了$\tilde{O}(n^{1.5})$运行时间，但我们对我们的特殊情况进行了更严格的分析，导致前面提到的更强的$\tilde{O}(\min\{m, \frac{n^2}{d} \})$时间。我们证明这种抽样方案的正确性使用了一个新的对应定理之间的削减和霍尔定理“的完美匹配的二分图，我们证明”。我们相信这个定理可能是独立的兴趣，作为另一个例子的应用，我们表明，在$n \times n$双随机矩阵与$m$非零条目的支持下，可以找到一个完美的匹配在预期的时间$\tilde{O}(m + n^{1.5})$。

英文摘要：

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 $\tilde{O}(m + n^{1.5})$.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

摘要：

英文摘要：

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