We present an O(m 10 7 ) = O(m 1.43 )-time 1 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 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 maximumcardinality 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.
For sparse graphs, the first improvement in 40 years is theÕ(m 10 7 ) time algorithm due to Madry REF .
Navigating Central Path with Electrical Flows: From Flows to Matchings, and Back
2013 IEEE 54th Annual Symposium on Foundations of Computer Science
2013 IEEE 54th Annual Symposium on Foundations of Computer Science
Mathematics Computer Science