Maximum Bipartite Matching in $n^{2+o(1)}$ Time via a Combinatorial Algorithm
Julia Chuzhoy, Sanjeev Khanna
TL;DR
This work presents a randomized combinatorial algorithm for Maximum Bipartite Matching that nearly matches the performance of IPM-based methods on dense graphs. The authors introduce a RouteAndCut framework, supported by the Approximate Topological Order (ATO) structure, MaintainCluster and ConnectToCenters subroutines, and a recursive bootstrapping between RouteAndCut and MaintainCluster across a parameter r. They leverage a modified multiplicative weights update (MWU) approach to reduce routing problems to decremental shortest-path problems, enabling n^{2+o(1)}-time performance with high probability. The technique hinges on maintaining expander-like clusters under online updates, and on an intricate layering data structure to connect all vertices to a central expander, ultimately yielding a randomized MBM algorithm that outperforms HK and matrix-multiplication methods in dense regimes. The approach extends to maximum vertex-capacitated flow with identical vertex capacities, demonstrating broad applicability of the combinatorial route-and-cut paradigm for flow-like problems.
Abstract
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.