# New Approximation Algorithms for Minimum Cycle Bases of Graphs

Ontology type: schema:Chapter      Open Access: True

### Chapter Info

DATE

2007-01-01

AUTHORS ABSTRACT

We consider the problem of computing an approximate minimum cycle basis of an undirected edge-weighted graph G with m edges and n vertices; the extension to directed graphs is also discussed. In this problem, a {0,1} incidence vector is associated with each cycle and the vector space over \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathbb{F}_2$\end{document} generated by these vectors is the cycle space of G. A set of cycles is called a cycle basis of G if it forms a basis for its cycle space. A cycle basis where the sum of the weights of the cycles is minimum is called a minimum cycle basis of G. Cycle bases of low weight are useful in a number of contexts, e.g. the analysis of electrical networks, structural engineering, chemistry, and surface reconstruction.We present two new algorithms to compute an approximate minimum cycle basis. For any integer k ≥ 1, we give (2k − 1)-approximation algorithms with expected running time O(kmn1 + 2/k + mn(1 + 1/k)(ω − 1)) and deterministic running time O( n3 + 2/k ), respectively. Here ω is the best exponent of matrix multiplication. It is presently known that ω < 2.376. Both algorithms are o( mω) for dense graphs. This is the first time that any algorithm which computes sparse cycle bases with a guarantee drops below the Θ(mω) bound.We also present a 2-approximation algorithm with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$O(m^{\omega}\sqrt{n\log n})$\end{document} expected running time, a linear time 2-approximation algorithm for planar graphs and an O(n3) time 2.42-approximation algorithm for the complete Euclidean graph in the plane. More... »

PAGES

512-523

### Book

TITLE

STACS 2007

ISBN

978-3-540-70917-6
978-3-540-70918-3

### Identifiers

URI

http://scigraph.springernature.com/pub.10.1007/978-3-540-70918-3_44

DOI

http://dx.doi.org/10.1007/978-3-540-70918-3_44

DIMENSIONS

https://app.dimensions.ai/details/publication/pub.1035485296

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