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

Abstract

We present an Õ(m <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">10/7</sup> ) = Õ(m <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1.43</sup> )-time <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sup> 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{√m, n <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2/3</sup> }) running time bound due to Even and Tarjan [16]. By well-known reductions, this also establishes an Õ(m <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">107</sup> )-time algorithm for the maximum-cardinality bipartite matching problem. That, in turn, gives an improvement over the celebrated O(m√n) running time bound of Hopcroft and Karp [25] whenever the input graph is sufficiently sparse. At a very high level, our results stem from acquiring a deeper understanding of interior-point methods - a powerful tool in convex optimization - in the context of flow problems, as well as, utilizing certain interplay between maximum flows and bipartite matchings.