Paper detail

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 in O(n^2) time to the problem of a logarithmic number of min-plus matrix multiplications of n-by-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 \tilde(O)(n^ωlog^3(nW/ε) / ε), where O(n^ω) is the time required for the classic n-by-n matrix multiplication and W is the maximum value of the weights.