Navigating Central Path with Electrical Flows: from Flows to Matchings, and Back

We present an $\tilde{O}(m^{10/7})=\tilde{O}(m^{1.43})$-time algorithm for the maximum s-t flow and the minimum s-t cut problems in directed graphs with unit capacities. This is the first improvement over the sparse-graph case of the long-standing $O(m \min(\sqrt{m},n^{2/3}))$ time bound due to Even and Tarjan [EvenT75]. By well-known reductions, this also establishes an $\tilde{O}(m^{10/7})$-time algorithm for the maximum-cardinality bipartite matching problem. That, in turn, gives an improvement over the celebrated celebrated $O(m \sqrt{n})$ time bound of Hopcroft and Karp [HK73] whenever the input graph is sufficiently sparse.