NDLI: Approximating the Girth

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Approximating the Girth

Content Provider	ACM Digital Library
Author	Tov, Roei Roditty, Liam
Copyright Year	2013
Abstract	This article considers the problem of computing a minimum weight cycle in weighted undirected graphs. Given a weighted undirected graph $\textit{G}$ = $(\textit{V},\textit{E},\textit{w}),$ let $\textit{C}$ be a minimum weight cycle of $\textit{G},$ let $\textit{w}(\textit{C})$ be the weight of $\textit{C},$ and let $w_{max}(C)$ be the weight of the maximum edge of $\textit{C}.$ We obtain three new approximation algorithms for the minimum weight cycle problem: (1) for integral weights from the range $[1,\textit{M}],$ an algorithm that reports a cycle of weight at most 4 $3\textit{w}(\textit{C})$ in $O(n^{2}$ log $\textit{n}(log$ $\textit{n}$ + log $\textit{M}))$ time; (2) For integral weights from the range $[1,\textit{M}],$ an algorithm that reports a cycle of weight at most $\textit{w}(\textit{C})$ + $w_{max}(C)$ in $O(n^{2}$ log $\textit{n}(log$ $\textit{n}$ + log $\textit{M}))$ time; (3) For nonnegative real edge weights, an algorithm that for any $\textit{ε}$ > 0 reports a cycle of weight at most (4 3 + $\textit{ε})\textit{w}(\textit{C})$ in $\textit{O}(1$ $\textit{ε}$ $n^{2}$ log $\textit{n}(log$ log $\textit{n}))$ 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 $[1,\textit{M}],$ implies a subcubic algorithm for computing all-pairs shortest paths in directed graphs with integral weights from the range $[™\textit{M},\textit{M}].$ 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 $\textit{n}(log$ $\textit{n}$ + log $\textit{M}))$ running time. They also posed, as an open problem, the question whether it is possible to obtain a subcubic algorithm with a $\textit{c}-approximation,$ where $\textit{c}$ < 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 ™ $\textit{ε})-approximation,$ where 0 < $\textit{ε}$ ≤ 1/3, implies a subcubic combinatorial algorithm for multiplying two boolean matrices.
Starting Page	1
Ending Page	13
Page Count	13
File Format	PDF
ISSN	15496325
e-ISSN	15496333
DOI	10.1145/2438645.2438647
Volume Number	9
Issue Number	2
Journal	ACM Transactions on Algorithms (TALG)
Language	English
Publisher	Association for Computing Machinery (ACM)
Publisher Date	2013-03-01
Publisher Place	New York
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Approximation Girth Graphs Minimum weight cycle
Content Type	Text
Resource Type	Article
Subject	Mathematics

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