Approximating the minimum cycle mean

We consider directed graphs where each edge is labeled with an integer weight and study the fundamental algorithmic question of computing the value of a cycle with minimum mean weight. Our contributions are twofold: (1) First we show that the algorithmic question is reducible to the problem of a logarithmic number of min-plus matrix multiplications of n×n-matrices, where n is the number of vertices of the graph. (2) Second, when the weights are nonnegative, we present the first (1+ϵ)-approximation algorithm for the problem and the running time of our algorithm is O˜(nωlog3(nW/ϵ)/ϵ),1 where O(nω) is the time required for the classicn×n-matrix multiplication and W is the maximum value of the weights. With an additional O(log(nW/ϵ)) factor in space a cycle with approximately optimal weight can be computed within the same time bound.