正则非二分图上的最大匹配

中文摘要 AI 辅助

阻塞流型最大匹配算法基于寻找最短增广路径的最大集合。它们在二分图[HK73, Din70, Kar73a, Kar73a]和非二分图[GT91, Gab17, Vaz24]上均以$O(m\sqrt{n})$时间运行，但当输入受到约束时，此时间界限可以得到改进。在本文中，我们考虑$d$-正则二分图和非二分图。先前的算法表明，在$d$-正则二分图中，完美匹配可以在确定性近线性时间[COS01]或高概率亚线性时间[GKK13]内计算。在$d$-正则非二分图上，可以在高概率下以亚线性时间$O(n \log n)$计算$(1-1/(d+1))$-近似[DH25]，因此可以在高概率下以$O(n^2)$时间计算最大匹配，这比正则图的最佳确定性算法[Yus13]（运行时间为$O(n^2 \log n)$）稍快。我们证明，任何基于寻找最短增广路径的阻塞流型最大匹配算法在$d$-正则图（包括二分和非二分）上运行时间为$O(n^2)$。在非二分图上，当$d = \omega(\sqrt{n})$时，这是对$O(n^2 \log n)$[Yus13]的渐近改进，也是对$O(m\sqrt{n})$[GT91, Gab17, Vaz24]的改进。它还通过使[DH25]的$O(n^2)$界限确定性化来改进[DH25]。然而，主要结论是无需新算法：“经典”匹配算法自动优于[Yus13, DH25]。我们还考虑了将我们的结果扩展到仅“近似正则”的图，即所有度数都位于指定范围$[d, \Delta]$内。

英文摘要

Blocking flow-type maximum matching algorithms are based on finding maximal sets of shortest augmenting paths. They run in $O(m\sqrt{n})$ time, on both bipartite [HK73, Din70, Kar73a, Kar73a] and nonbipartite graphs [GT91, Gab17, Vaz24], but this time bound can be improved if the input is constrained. In this paper we consider $d$-regular bipartite and nonbipartite graphs. Previous algorithms show that a perfect matching in $d$-regular bipartite graphs can be computed in near-linear time deterministically [COS01] or sublinear time with high probability [GKK13]. On $d$-regular non-bipartite graphs, a $(1-1/(d+1))$-approximation can be computed in sublinear time $O(n \log n)$ with high probability [DH25], and hence a maximum matching can be computed in $O(n^2)$ time, w.h.p., which is slightly faster than the best deterministic algorithm for regular graphs [Yus13], running in O(n^2 log n) time. We prove that any blocking flow-type maximum matching algorithm based on finding shortest augmenting paths runs in $O(n^2)$ time on d-regular graphs, both bipartite and nonbipartite. On nonbipartite graphs this is an asymptotic improvement over $O(n^2 \log n)$ [Yus13] and an improvement over $O(m\sqrt{n})$ [GT91, Gab17, Vaz24] when $d = ω(\sqrt{n})$. It also improves [DH25] by making its $O(n^2)$ bound deterministic. However, the main take-away message is that no new algorithms are needed: the "classic" matching algorithms automatically outperform [Yus13, DH25]. We also consider extensions of our results to graphs that are only "nearly regular," meaning that their degrees all lie within a specified range, $[d, Δ]$.

发表机构

• Rochester Institute of Technology（罗切斯特理工学院）
• University at Buffalo（布法罗大学）
• University of Michigan（密歇根大学）

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