New approximation algorithms for minimum cycle bases of graphs

We consider the problem of computing an approximate minimum cycle basis of an undirected non-negative edge-weighted graph G with m edges and n vertices; the extension to directed graphs is also discussed. 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. Cycle bases of low weight are useful in a number of contexts, e.g. the analysis of electrical networks, structural engineering, chemistry, and surface reconstruction. We present two new algorithms to compute an approximate minimum cycle basis. For any integer k ≥ 1, we give (2k−1)-approximation algorithms with expected running time O(kmn 1+2/k + mn (1+1/k)(ω−1) ) and deterministic running time O(n 3+2/k), respectively. Here ω is the best exponent of matrix multiplication. It is presently known that ω < 2.376. Both algorithms are o(m ω) for dense graphs. This is the first time that any algorithm which computes sparse cycle bases with a guarantee drops below the Θ(m ω) bound. We also present a 2-approximation algorithm with expected running time O(m ω √ nlogn), a linear time 2-approximation algorithm for planar graphs and an O(n 3) time 2.42-approximation algorithm for the complete Euclidean graph in the plane