Global minimum cuts in surface embedded graphs

We give a deterministic algorithm to find the minimum cut in a surface-embedded graph in near-linear time. Given an undirected graph embedded on an orientable surface of genus g, our algorithm computes the minimum cut in gO(g) nloglog n time, matching the running time of the fastest algorithm known for planar graphs, due to Ł ˛acki and Sankowski, for any constant g. Indeed, our algorithm calls Ł ˛acki and Sankowski’s recent O(nloglog n) time planar algorithm as a subroutine. Previously, the best time bounds known for this problem followed from two algorithms for general sparse graphs: a randomized algorithm of Karger that runs in O(nlog3 n) time and succeeds with high probability, and a deterministic algorithm of Nagamochi and Ibaraki that runs in O(n2 log n) time. We can also achieve a deterministic gO(g) n2 loglog n time bound by repeatedly applying the best known algorithm for minimum (s, t)-cuts in surface graphs. The bulk of our work focuses on the case where the dual of t...

• Type: Journal
• Year: 2012
• Where: SODA
• Authors: Jeff Erickson, Kyle Fox, Amir Nayyeri
• Keywords: Algorithms, Deterministic Algorithm, Randomized Algorithm, SODA 2012, Sparse Graphs