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 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 ε > 0 reports a cycle of weight at most in 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 < 2. The current paper answers this question in the affirmative, by presenting an algorithm with 4/3-approximation and the same running time. Surprisingly, the approximation factor of 4/3 is not accidental. We show using the new result of Vassilevska Williams and Williams [WW10] that a subcubic combinatorial algorithm with (4/3 − ε)-approximation, where 0 < ε ≤ 1/3, implies a subcubic combinatorial algorithm for multiplying two boolean matrices.