Maximum Bipartite Matching in 𝑛^2+𝑜(1) Time via a Combinatorial Algorithm

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 √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, that builds on continuous techniques based on interior-point methods, 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. One reason to explore combinatorial algorithms is that they are often more transparent than their continuous counterparts, and that the tools and techniques developed for such algorithms may be useful in other settings, including, for example, developing faster algorithms for maximum matching in general graphs. A recent work of Chuzhoy and Khanna (2024) made progress on this question by giving a combinatorial Õ(m^1/3n^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. Similar to the classical algorithms for MBM and the approach used in the work of Chuzhoy and Khanna (2024), our algorithm is based on iterative augmentation of a current matching using augmenting paths in the corresponding (directed) residual flow network. Our main contribution is a recursive algorithm that exploits the special structure of the resulting flow problem to recover an Ω(1/log² n)-fraction of the remaining augmentations in n^2+o(1) time. Finally, we obtain a randomized n^2+o(1)-time 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.