An Õ(m ² n) Randomized Algorithm to Compute a Minimum Cycle Basis of a Directed Graph

Abstract

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 ${\mathbb Q}$ 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 $\tilde{O}(m{^{\omega+1}}n)$ time (where ω < 2.376 is the exponent of matrix multiplication). If one allows randomization, then an Õ(m ³ n) algorithm is known for this problem. In this paper we present a simple Õ(m ² n) 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 ${\mathbb F}_{2}$ generated by these vectors is the cycle space of the graph. The fastest known algorithm for computing a minimum cycle basis in an undirected graph runs in O(m ² n + mn ²log n) time and our randomized algorithm for directed graphs almost matches this running time.