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An (O)over-tilde (m(2)n) Algorithm for Minimum Cycle Basis of Graphs

Kavitha, Telikepalli and Mehlhorn, Kurt and Michail, Dimitrios and Paluch, Katarzyna E (2008) An (O)over-tilde (m(2)n) Algorithm for Minimum Cycle Basis of Graphs. In: Algorithmica, 52 (3). pp. 333-349.

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We consider the problem of computing a minimum cycle basis of an undirected non-negative edge-weighted graph G with m edges and n vertices. In this problem, a {0,1} incidence vector is associated with each cycle and the vector space over $\mathbb{F}_{2}$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. Minimum cycle basis are useful in a number of contexts, e.g. the analysis of electrical networks and structural engineering. The previous best algorithm for computing a minimum cycle basis has running time O(m ω n), where ω is the best exponent of matrix multiplication. It is presently known that ω<2.376. We exhibit an O(m 2 n+mn 2log n) algorithm. When the edge weights are integers, we have an O(m 2 n) algorithm. For unweighted graphs which are reasonably dense, our algorithm runs in O(m ω ) time. For any ε>0, we also design an 1+ε approximation algorithm. The running time of this algorithm is O((m ω /ε)log (W/ε)) for reasonably dense graphs, where W is the largest edge weight.

Item Type: Journal Article
Publication: Algorithmica
Publisher: Springer
Additional Information: Copyright of this article belongs to Springer.
Keywords: Cycle basis;Cycle space;Matrix multiplication;Polynomial algorithms.
Department/Centre: Division of Electrical Sciences > Computer Science & Automation
Date Deposited: 01 May 2009 03:02
Last Modified: 19 Sep 2010 04:57
URI: http://eprints.iisc.ac.in/id/eprint/17553

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