Minimum Cuts in Directed Graphs via Partial Sparsification
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We give an algorithm to find a minimum cut in an edge-weighted directed graph with n vertices and m edges in Õ(n·max(m^2/3, n)) time. This improves on the 30 year old bound of Õ(nm) obtained by Hao and Orlin for this problem. Our main technique is to reduce the directed mincut problem to Õ(min(n/m^1/3, √(n))) calls of any maxflow subroutine. Using state-of-the-art maxflow algorithms, this yields the above running time. Our techniques also yield fast approximation algorithms for finding minimum cuts in directed graphs. For both edge and vertex weighted graphs, we give (1+ϵ)-approximation algorithms that run in Õ(n^2 / ϵ^2) time.
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