A Faster Algorithm for Finding the Minimum Cut in a Directed Graph

We consider the problem of finding the minimum capacity cut in a directed network G with n nodes. This problem has applications to network reliability and survivability and is useful in subroutines for other network optimization problems. One can use a maximum flow problem to find a minimum cut separating a designated source node s from a designated sink node t, and by varying the sink node one can find a minimum cut in G as a sequence of at most 2n-2 maximum flow problems. We then show how to reduce the running time of these 2n-2 maximum flow algorithms to the running time for solving a single maximum flow problem. The resulting running time is O(nm log(n 2 /m)) for finding the minimum cut in either a directed or an undirected network.