An Õ(n2) algorithm for minimum cuts

A minimum cut is a set of edges of minimum weight whose removal disconnects a given graph. Minimum cut algorithms historically applied duality with maximum flows and thus had the same 0 (inn) running time as maximum flow algorithms. More recent algorithms which are not based on maximum flows also require fl (inn) time. In this paper, we present the first algorithm that breaks the tl(mn) “max-flow barrier” for finding minimum cuts in weighted undirected graphs. We give a strongly polynomial randomized algorithm which finds a minimum cut with high probability in 0(n2 log3 n) time. This suggests that the rein-cut problem might be fundamentally easier to solve than the maximum flow problem. Our algorithm can be implemented in 72JUC using only nz processors—this is the first efficient 7UfC algorithm for the rein-cut problem. Our algorithm is simple and uses no complicated data structures.