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 x n-matrices, where n is the number of vertices of the graph. (2) Second, when the weights are nonnegative, we present the first (1 + is an element of)-approximation algorithm for the problem and the running time of our algorithm is (O) over tilde (n(omega) log(3) (nW/is an element of)/is an element of),(1) where O(n(omega)) is the time required for the classic n x n-matrix multiplication and W is the maximum value of the weights. With an additional O(log(nW/is an element of)) factor in space a cycle with approximately optimal weight can be computed within the same time bound. (C) 2014 Elsevier B.V. All rights reserved.