Item

Abstract

In this paper we consider the problem of computing a minimum cycle basis in a graph $G$ with $m$ edges and $n$ vertices. The edges of $G$ have non-negative weights on them. The previous best result for this problem was an $O(m^{\omega}n)$ algorithm, where $\omega$ is the best exponent of matrix multiplication. It is presently known that $\omega < 2.376$. We obtain an $O(m^2n + mn^2\log n)$ algorithm for this problem. Our algorithm also uses fast matrix multiplication. When the edge weights are integers, we have an $O(m^2n)$ algorithm. For unweighted graphs which are reasonably dense, our algorithm runs in $O(m^{\omega})$ time. For any $\epsilon > 0$, we also design a $1+\epsilon$ approximation algorithm to compute a cycle basis which is at most $1+\epsilon$ times the weight of a minimum cycle basis. The running time of this algorithm is $O(\frac{m^{\omega}}{\epsilon}\log(W/{\epsilon}))$ for reasonably dense graphs, where $W$ is the largest edge weight.