A Faster Deterministic Algorithm for Minimum Cycle Basis in Directed Graphs

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 f 1;0;1g 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 set of cycles is called a cycle basis of G if it forms a basis for this vector space. We seek a cycle basis where the sum of weights of the cycles is minimum. The current fastest algorithm for computing a minimum cycle basis in a directed graph with m edges and n vertices runs in Õ(mω+1n) time (where ω < 2:376 is the exponent of matrix multiplication). Here we present an O(m3n+m2n2 logn) algorithm. This algorithm is obtained by using fast matrix multiplication over rings and an efficient extension of Dijkstra’s algorithm to compute a shortest cycle in G whose dot product with a function on its edge set is non-zero. Indian Institute of Science, Bangalore. Email: [email protected] †Indian Institute of Science, Bangalore. Email: [email protected]. This work was done while visiting the Max-PlanckInstitut für Informatik, Saarbrücken. ‡Max-Planck-Institut für Informatik, Saarbrücken. Email: [email protected]