A Faster Algorithm for Minimum Cycle Basis of Graphs

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 ^ω n) algorithm, where ω is the best exponent of matrix multiplication. It is presently known that ω < 2.376. We obtain an O(m ² n + mn ²log n) algorithm for this problem. Our algorithm also uses fast matrix multiplication. 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 a 1+ε approximation algorithm to compute a cycle basis which is at most 1+ε 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.