Minimum cuts and shortest homologous cycles

We describe the first algorithms to compute minimum cuts in surface-embedded graphs in near-linear time. Given an undirected graph embedded on an orientable surface of genus g, with two specified vertices s and t, our algorithm computes a minimum (s, t)-cut in gO(g) nlog n time. Except for the special case of planar graphs, for which O(nlog n)-time algorithms have been known for more than 20 years, the best previous time bounds for finding minimum cuts in embedded graphs follow from algorithms for general sparse graphs. A slight generalization of our minimum-cut algorithm computes a minimum-cost subgraph in every 2-homology class. We also prove that finding a minimum-cost subgraph homologous to a single input cycle is NP-hard. Categories and Subject Descriptors: F.2.2 [Analysis of Algorithms and Problem Complexity]: Nonnumerical Algorithms and Problems—Computations on discrete structures; G.2.2 [Discrete Mathematics]: Graph theory—Graph algorithms General Terms: Algorithms, Pe...

• Type: Conference
• Year: 2009
• Where: COMPGEOM
• Authors: Erin W. Chambers, Jeff Erickson, Amir Nayyeri
• Keywords: Algorithms, COMPGEOM 2009, Discrete Geometry, Minimum Cuts, Minimum-cost Subgraph