Counting and Sampling Minimum Cuts in Genus g Graphs

Let \(G\) be a directed graph with \(n\) vertices embedded on an orientable surface of genus \(g\) with two designated vertices \(s\) and \(t\) . We show that computing the number of minimum \((s,t)\) -cuts in \(G\) is fixed-parameter tractable in \(g\) . Specifically, we give a \(2^{O(g)} n^2\) time algorithm for this problem. Our algorithm requires counting sets of cycles in a particular integer homology class. That we can count these cycles is an interesting result in itself as there are no prior results that are fixed-parameter tractable and deal directly with integer homology. We also describe an algorithm which, after running our algorithm to count minimum cuts once, can sample an \((s,t)\) -minimum cut uniformly at random in \(O(n \log n)\) time per sample.