Madry, Aleksander2014-06-022014-06-022014-06-02201310.1109/Focs.2013.35https://infoscience.epfl.ch/handle/20.500.14299/103905WOS:000330672400027We present an O(m^10/7) = 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{m^1/2, n^2/3}) running time bound due to Even and Tarjan [16]. By well-known reductions, this also establishes an O(m^10/7)-time algorithm for the maximum-cardinality bipartite matching problem. That, in turn, gives an improvement over the celebrated O(mn^1/2) 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.maximum flow problemminimum s-t cut problembipartite matchingselectrical flowsinterior-point methodscentral pathLaplacian linear systemstext::conference output::conference proceedings::conference paper