An $\tilde{O}(m^{2}n)$ Algorithm for Minimum Cycle Basis of Graphs
Telikepalli Kavitha, Kurt Mehlhorn, Dimitrios Michail, Katarzyna E. Paluch · Algorithmica · 2007
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 ω0, 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.