Maximum bipartite matching (MBM) is a fundamental problem in combinatorial optimization with a long and rich history. A classic result of Hopcroft and Karp (1973) provides an $O(m \sqrt{n})$-time algorithm for the problem, where $n$ and $m$ are the number of vertices and edges in the input graph, respectively. For dense graphs, an approach based on fast matrix multiplication achieves a running time of $O(n^{2.371})$. For several decades, these results represented state-of-the-art algorithms, until, in 2013, Madry introduced a powerful new approach for solving MBM using continuous optimization techniques. This line of research led to several spectacular results, culminating in a breakthrough $m^{1+o(1)}$-time algorithm for min-cost flow, that implies an $m^{1+o(1)}$-time algorithm for MBM as well. These striking advances naturally raise the question of whether combinatorial algorithms can match the performance of the algorithms that are based on continuous techniques for MBM. A recent work of the authors (2024) made progress on this question by giving a combinatorial $\tilde{O}(m^{1/3}n^{5/3})$-time algorithm for MBM, thus outperforming both the Hopcroft-Karp algorithm and matrix multiplication based approaches, on sufficiently dense graphs. Still, a large gap remains between the running time of their algorithm and the almost linear-time achievable by algorithms based on continuous techniques. In this work, we take another step towards narrowing this gap, and present a randomized $n^{2+o(1)}$-time combinatorial algorithm for MBM. Thus in dense graphs, our algorithm essentially matches the performance of algorithms that are based on continuous methods. We also obtain a randomized $n^{2+o(1)}$-time combinatorial algorithm for maximum vertex-capacitated $s$-$t$ flow in directed graphs when all vertex capacities are identical, using a standard reduction from this problem to MBM.

翻译：最大二分图匹配（MBM）是组合优化中的一个基础问题，具有悠久而丰富的历史。Hopcroft 和 Karp（1973）的经典结果为该问题提供了一个 $O(m \sqrt{n})$ 时间的算法，其中 $n$ 和 $m$ 分别是输入图中的顶点数和边数。对于稠密图，一种基于快速矩阵乘法的方法实现了 $O(n^{2.371})$ 的运行时间。几十年来，这些结果代表了最先进的算法，直到 2013 年，Madry 引入了一种利用连续优化技术求解 MBM 的强大新方法。这一研究方向取得了多项显著成果，最终催生了求解最小费用流问题的突破性 $m^{1+o(1)}$ 时间算法，该算法也意味着存在 $m^{1+o(1)}$ 时间的 MBM 算法。这些引人注目的进展自然引出一个问题：对于 MBM 问题，组合算法能否达到基于连续技术的算法的性能？作者近期的工作（2024）在这一问题上取得了进展，给出了一个组合的 $\tilde{O}(m^{1/3}n^{5/3})$ 时间 MBM 算法，从而在足够稠密的图上超越了 Hopcroft-Karp 算法和基于矩阵乘法的方法。然而，他们的算法运行时间与基于连续技术的算法所能达到的几乎线性时间之间仍存在巨大差距。在本工作中，我们朝着缩小这一差距又迈进了一步，提出了一个随机化的 $n^{2+o(1)}$ 时间组合算法用于求解 MBM。因此，在稠密图中，我们的算法本质上匹配了基于连续方法的算法的性能。利用从该问题到 MBM 的标准规约，我们还得到了一个随机化的 $n^{2+o(1)}$ 时间组合算法，用于求解有向图中所有顶点容量相同的最大顶点容量 $s$-$t$ 流问题。