## Minimum Cuts and Shortest Non-Separating Cycles via Homology Covers (2011)

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Venue: | SYMPOSIUM ON DISCRETE ALGORITHMS |

Citations: | 9 - 3 self |

### BibTeX

@INPROCEEDINGS{Erickson11minimumcuts,

author = {Jeff Erickson and Amir Nayyeri},

title = {Minimum Cuts and Shortest Non-Separating Cycles via Homology Covers},

booktitle = {SYMPOSIUM ON DISCRETE ALGORITHMS},

year = {2011},

publisher = {}

}

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### Abstract

Let G be a directed graph with weighted edges, embedded on a surface of genus g with b boundaries. We describe an algorithm to compute the shortest directed cycle in G in any given � 2-homology class in 2 O(g+b) n log n time; this problem is NP-hard even for undirected graphs. We also present two applications of our algorithm. The first is an algorithm to compute the shortest non-separating directed cycle in G in 2 O(g) n log n time, improving the recent algorithm of Cabello et al. [SOCG 2010] for all g = o(log n). The second is a combinatorial algorithm to compute minimum (s, t)-cuts in undirected surface graphs in 2 O(g) n log n time, improving an algorithm of Chambers et al. [SOCG 2009] for all positive g. Unlike earlier algorithms for surface graphs that construct and search finite portions of the universal cover, our algorithms use another canonical covering space, called the Z 2-homology cover.