Bipartite Matching in Nearly-linear Time on Moderately Dense Graphs

We present an $\tilde{O}(m+n^{1.5})$-time randomized algorithm for maximum cardinality bipartite matching and related problems (e.g. transshipment, negative-weight shortest paths, and optimal transport) on m-edge, n-node graphs. For maximum cardinality bipartite matching on moderately dense graphs, i.e. $m=\Omega(n^{1.5})$, our algorithm runs in time nearly linear in the input size and constitutes the first improvement over the classic $O(m\sqrt{n})$-time [Dinic 1970; Hopcroft-Karp 1971; Karzanov 1973] and $\widetilde{O}(n^{\omega})$-time algorithms [Ibarra-Moran 1981] (where currently $\omega\approx 2.373$). On sparser graphs, i.e. when $m=n^{9/8+\delta}$ for any constant $\delta > 0$, our result improves upon the recent advances of [Madry 2013] and [Liu-Sidford 2020b, 2020a] which achieve an $\widetilde{O}(m^{4/3+o(1)})$ runtime. We obtain these results by combining and advancing recent lines of research in interior point methods (IPMs) and dynamic graph algorithms. First, we simplify and improve the IPM of [v.d.Brand-Lee-Sidford-Song 2020], providing a general primal-dual IPM framework and new sampling-based techniques for handling infeasibility induced by approximate linear system solvers. Second, we provide a simple sublinear-time algorithm for detecting and sampling high-energy edges in electric flows on expanders and show that when combined with recent advances in dynamic expander decompositions, this yields efficient data structures for maintaining the iterates of both [v.d.Brand et al.] and our new IPMs. Combining this general machinery yields a simpler $\widetilde{O}(n\sqrt{m})$ time algorithm for matching based on the logarithmic barrier function, and our state-of-the-art $\widetilde{O}(m+n^{1.5})$ time algorithm for matching based on the [Lee-Sidford 2014] barrier (as regularized in [v.d.Brand et al.]).