An Õ(m2n) Randomized Algorithm to Compute a Minimum Cycle Basis of a Directed Graph

We consider the problem of computing a minimum cycle basis in a directed graph G. The input to this problem is a directed graph whose arcs have positive weights. In this problem a 1 0 1 incidence vector is associated with each cycle and the vector space over 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 weights of the cycles is minimum is called a minimum cycle basis of G. The current fastest algorithm for computing a minimum cycle basis in a directed graph with m arcs and n vertices runs in Õ mω 1n time (where ω 2 376 is the exponent of matrix multiplication). If one allows randomization, then an Õ m3n algorithm is known for this problem. In this paper we present a simple Õ m2n randomized algorithm for this problem. The problem of computing a minimum cycle basis in an undirected graph has been well-studied. In this problem a 0 1 incidence vector is associated with each cycle and the vector space over 2 generated by these vectors is the cycle space of the graph. It is not known if an efficient algorithm for undirected graphs automatically translates to an efficient algorithm for directed graphs. The fastest known algorithm for computing a minimum cycle basis in an undirected graph runs in O m2n mn2 logn time and our randomized algorithm for directed graphs almost matches this running time.