Bipartite Matching in Nearly-linear Time on Moderately Dense Graphs

We present an ~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=Ω(n ^1.5 ), our algorithm runs in time nearly linear in the input size and constitutes the first improvement over the classic O(m√n)-time [Dinic 1970; Hopcroft-Karp 1971; Karzanov 1973] and ~O(n ^ω )-time algorithms [Ibarra-Moran 1981] (where currently ω ≈ 2.373). On sparser graphs, i.e. when m=n ^9/8+δ for any constant , our result improves upon the recent advances of [Madry 2013] and [Liu-Sidford 2020b, 2020a] which achieve an ~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 ~O(n√m) time algorithm for matching based on the logarithmic barrier function, and our state-of-the-art ~O(m+n ^1.5 ) time algorithm for matching based on the [Lee-Sidford 2014] barrier (as regularized in [v.d.Brand et al.]).