A Faster Deterministic Algorithm for Minimum Cycle Bases in Directed Graphs

Abstract

We consider the problem of computing a minimum cycle basis in a directed graph. The input to this problem is a directed graph G whose edges have non-negative weights. A cycle in this graph is actually a cycle in the underlying undirected graph with edges traversable in both directions. A {–1,0,1} edge incidence vector is associated with each cycle: edges traversed by the cycle in the right direction get 1 and edges traversed in the opposite direction get -1. The vector space over ℚ generated by these vectors is the cycle space of G. A minimum cycle basis is a set of cycles of minimum weight that span the cycle space of G. The current fastest algorithm for computing a minimum cycle basis in a directed graph with m edges and n vertices runs in $\tilde{O}(m^{\omega+1}n)$ time (where ω< 2.376 is the exponent of matrix multiplication). Here we present an O(m ³ n + m ² n ²logn) algorithm. We also slightly improve the running time of the current fastest randomized algorithm from O(m ² nlogn) to O(m ² n + mn ² logn).