Optimal Pants Decompositions and Shortest Homotopic Cycles on an Orientable Surface

Abstract

A pants decomposition of a compact orientable surface M is a set of disjoint simple cycles which cuts M into pairs of pants, i.e., spheres with three boundaries. Assuming M is a polyhedral surface, with weighted vertex-edge graph G, we consider combinatorial pants decompositions: the cycles are closed walks in G that may overlap but do not cross. We give an algorithm which, given a pants decomposition, computes a homotopic pants decomposition in which each cycle is a shortest cycle in its homotopy class. In particular, the resulting decomposition is optimal (as short as possible among all homotopic pants decompositions), and any optimal pants decomposition is made of shortest homotopic cycles. Our algorithm is polynomial in the complexity of the input and in the longest-to-shortest edge ratio of G. The same algorithm can be applied, given a simple cycle C, to compute a shortest cycle homotopic to C which is itself simple.