# An $\~{O}(m^2n)$ randomized algorithm to compute a minimum cycle basis of a directed graph

Kavitha, Telikepalli (2005) An $\~{O}(m^2n)$ randomized algorithm to compute a minimum cycle basis of a directed graph. In: ICALP, Jul 11-15 2005, Lisbon, Portugal, pp. 273-284.

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## Abstract

We consider the problem of computing a minimum cycle basis in a directed graph G. The input to this problem is a directed graph whose arcs have positive weights. In this problem a {−1, 0, 1} incidence vector is associated with each cycle and the vector space over $\!Q$ 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 weights of the cycles is minimum is called a minimum cycle basis of G. The current fastest algorithm for computing a minimum cycle basis in a directed graph with m arcs and n vertices runs in $\~{O}(m^{\omega+1}n)$ time (where $\omega$ < 2.376 is the exponent of matrix multiplication). If one allows randomization, then an $\~{O}(m^3n)$ algorithm is known for this problem. In this paper we present a simple $\~{O}(m^2n)$ randomized algorithm for this problem. The problem of computing a minimum cycle basis in an undirected graph has been well-studied. In this problem a {0, 1} incidence vector is associated with each cycle and the vector space over $\!F_2$ generated by these vectors is the cycle space of the graph. The fastest known algorithm for computing a minimum cycle basis in an undirected graph runs in $O(m^2n + mn^2 log n)$ time and our randomized algorithm for directed graphs almost matches this running time.

Item Type: Conference Paper Copyright of this article belongs to Springer-Verlag Berlin Heidelberg Division of Electrical Sciences > Computer Science & Automation (Formerly, School of Automation) 28 Feb 2008 19 Sep 2010 04:42 http://eprints.iisc.ernet.in/id/eprint/13079