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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## Abstract

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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