An Õ(mn) Algorithm for Minimum Cycle Basis of Graphs∗

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 F2 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(mn), where ω is the best exponent of matrix multiplication. It is presently known that ω < 2.376. We exhibit an O(mn + mn logn) algorithm. When the edge weights are integers, we have an O(mn) algorithm. For unweighted graphs which are reasonably dense, our algorithm runs in O(m) time. For any > 0, we also design an 1 + approximation algorithm. The running time of this algorithm is O((m/ ) log(W/ )) for reasonably dense graphs, where W is the largest edge weight.