Approximating the Girth

This article considers the problem of computing a minimum weight cycle in weighted undirected graphs. Given a weighted undirected graph <i>G</i> = (<i>V</i>,<i>E</i>,<i>w</i>), let <i>C</i> be a minimum weight cycle of <i>G</i>, let <i>w</i>(<i>C</i>) be the weight of <i>C</i>, and let <i>w</i>_max(<i>C</i>) be the weight of the maximum edge of <i>C</i>. We obtain three new approximation algorithms for the minimum weight cycle problem: (1) for integral weights from the range [1,<i>M</i>], an algorithm that reports a cycle of weight at most 4 3<i>w</i>(<i>C</i>) in <i>O</i>(<i>n</i>² log <i>n</i>(log <i>n</i> + log <i>M</i>)) time; (2) For integral weights from the range [1,<i>M</i>], an algorithm that reports a cycle of weight at most <i>w</i>(<i>C</i>) + <i>w</i>_max(<i>C</i>) in <i>O</i>(<i>n</i>² log <i>n</i>(log <i>n</i> + log <i>M</i>)) time; (3) For nonnegative real edge weights, an algorithm that for any <i>ε</i> > 0 reports a cycle of weight at most (4 3 + <i>ε</i>)<i>w</i>(<i>C</i>) in <i>O</i>(1 <i>ε</i> <i>n</i>² log <i>n</i>(log log <i>n</i>)) 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,<i>M</i>], implies a subcubic algorithm for computing all-pairs shortest paths in directed graphs with integral weights from the range [−<i>M</i>,<i>M</i>]. 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 <i>O</i>(<i>n</i>² log <i>n</i>(log <i>n</i> + log <i>M</i>)) running time. They also posed, as an open problem, the question whether it is possible to obtain a subcubic algorithm with a <i>c</i>-approximation, where <i>c</i> < 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 − <i>ε</i>)-approximation, where 0 < <i>ε</i> ≤ 1/3, implies a subcubic combinatorial algorithm for multiplying two boolean matrices.