Minimum cuts and shortest homologous cycles

Authors:
Erin W. Chambers;Jeff Erickson;Amir Nayyeri
Affiliations:
Saint Louis University, St. Louis, MO, USA;University of Illinois, Urbana, IL, USA;University of Illinois, Urbana, IL, USA
Venue:
Proceedings of the twenty-fifth annual symposium on Computational geometry
Year:
2009

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Abstract

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) n log n time. Except for the special case of planar graphs, for which O(n log 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 Z2-homology class. We also prove that finding a minimum-cost subgraph homologous to a single input cycle is {NP}-hard.