Optimal pants decompositions and shortest homotopic cycles on an orientable surface

We consider the problem of finding a shortest cycle (freely) homotopic to a given simple cycle on a compact, orientable surface. For this purpose, we use a pants decomposition of the surface: a set of disjoint simple cycles that cut the surface into pairs of pants (spheres with three holes). We solve this problem in a framework where the cycles are closed walks on the vertex-edge graph of a combinatorial surface that may overlap but do not cross. We give an algorithm that transforms an input pants decomposition into another homotopic pants decomposition that is optimal: each cycle is as short as possible in its homotopy class. As a consequence, finding a shortest cycle homotopic to a given simple cycle amounts to extending the cycle into a pants decomposition and to optimizing it: the resulting pants decomposition contains the desired cycle. We describe two algorithms for extending a cycle to a pants decomposition. All algorithms in this article are polynomial, assuming uniformity of the weights of the vertex-edge graph of the surface.