Approximating the Girth

This article 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 w_max(C) be the weight of the maximum 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) + w_max(C) in O(n² log n(log n + log M)) time; (3) For nonnegative real edge weights, an algorithm that for any ε > 0 reports a cycle of weight at most (4 3 + ε)w(C) in O(1 ε n² log n(log log n)) time.

In a recent breakthrough, Williams and Williams [2010] 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 [2009] 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 article 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 Williams and Williams [2010], 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.