Approximating the girth

Liam Roditty, Roei Tov

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

6 Scopus citations

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 w max(C) be the weight of the maximal edge of C. We obtain three new approximation algorithms for the minimum weight cycle problem: 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 ε > 0 reports a cycle of weight at most (4/3 + ε)w(C) in O(1/εn 2log 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 < 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 < e ≤ 1/3, implies a subcubic combinatorial algorithm for multiplying two boolean matrices.

Original languageEnglish
Title of host publicationProceedings of the 22nd Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2011
PublisherAssociation for Computing Machinery
Pages1446-1454
Number of pages9
ISBN (Print)9780898719932
DOIs
StatePublished - 2011

Publication series

NameProceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms

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