The maximum bipartite matching problem is among the most fundamental and well-studied problems in combinatorial optimization. A beautiful and celebrated combinatorial algorithm of Hopcroft and Karp (1973) shows that maximum bipartite matching can be solved in $O(m \sqrt{n})$ time on a graph with $n$ vertices and $m$ edges. For the case of very dense graphs, a fast matrix multiplication based approach gives a running time of $O(n^{2.371})$. These results represented the fastest known algorithms for the problem until 2013, when Madry introduced a new approach based on continuous techniques achieving much faster runtime in sparse graphs. This line of research has culminated in a spectacular recent breakthrough due to Chen et al. (2022) that gives an $m^{1+o(1)}$ time algorithm for maximum bipartite matching (and more generally, for min cost flows). This raises a natural question: are continuous techniques essential to obtaining fast algorithms for the bipartite matching problem? Our work makes progress on this question by presenting a new, purely combinatorial algorithm for bipartite matching, that runs in $\tilde{O}(m^{1/3}n^{5/3})$ time, and hence outperforms both Hopcroft-Karp and the fast matrix multiplication based algorithms on moderately dense graphs. Using a standard reduction, we also obtain an $\tilde{O}(m^{1/3}n^{5/3})$ time deterministic algorithm for maximum vertex-capacitated $s$-$t$ flow in directed graphs when all vertex capacities are identical.

翻译：最大二分匹配问题是组合优化中最基础且研究最深入的问题之一。Hopcroft和Karp（1973）提出的优美而著名的组合算法表明，在包含n个顶点和m条边的图上，最大二分匹配可以在$O(m \sqrt{n})$时间内求解。对于非常稠密的图，基于快速矩阵乘法的方法可实现$O(n^{2.371})$的运行时间。这些结果曾是该问题已知最快的算法，直至2013年Madry引入基于连续技术的新方法，在稀疏图上实现了更快的运行速度。这一研究方向最终在Chen等人（2022年）的突破性成果中达到顶峰——他们提出了$m^{1+o(1)}$时间复杂度的最大二分匹配算法（更一般地，适用于最小费用流问题）。这引发了一个自然问题：连续技术对于获得二分匹配问题的快速算法是否必不可少？我们的工作在这一问题上取得了进展，提出了一种新的纯组合二分匹配算法，运行时间为$\tilde{O}(m^{1/3}n^{5/3})$，因此在中等稠密图上优于Hopcroft-Karp算法和基于快速矩阵乘法的方法。通过标准归约，我们还得到了一个$\tilde{O}(m^{1/3}n^{5/3})$时间的确定性算法，用于所有顶点容量相同的有向图中最大顶点容量$s$-$t$流问题。