Approximating the Girth

Abstract

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 3 w ( C ) in O ( n 2 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 ( n 2 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 n 2 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 ( n 2 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.