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.]).