Approximating the girth

Abstract

This paper considers the problem of computing a minimum weight cycle in weighted undirected graphs. Given a weighted undirected graph G(V,E,w), let C be a minimum weight cycle of G, let w(C) be the weight of C and let wmax (C) be the weight of the maximal edge of C. We obtain three new approximation algorithms for the minimum weight cycle problem:1. For integral weights from the range [1, M] an algorithm that reports a cycle of weight at most 4/3w(C) in O(n2 log n(log n + log M)) time.2. For integral weights from the range [1, M] an algorithm that reports a cycle of weight at most w(C) + wmax (C) in O(n2 log n(log n + log M)) time.3. For non-negative real edge weights an algorithm that for any e > 0 reports a cycle of weight at most (4/3 + e)w(C) in O(1/e n2 log n(log log n)) time.In a recent breakthrough Vassilevska Williams and Williams [WW10] showed that a subcubic algorithm that computes the exact minimum weight cycle in undirected graphs with integral weights from the range [1, M] implies a subcubic algorithm for computing all-pairs shortest paths in directed graphs with integral weights from the range [− M, M]. This implies that in order to get a subcubic algorithm for computing a minimum weight cycle we have to relax the problem and to consider an approximated solution.Lingas and Lundell [LL09] were the first to consider approximation in the context of minimum weight cycle in weighted graphs. They presented a 2-approximation algorithm for integral weights with O(n2 log n(log n + log M)) running time. They also posed as an open problem the question whether it is possible to obtain a subcubic algorithm with a c-approximation, where c