Approximating the minimum cycle mean

TL;DR

This paper investigates algorithms for finding the minimum cycle mean in weighted directed graphs, providing a reduction to min-plus matrix multiplication and a new approximation algorithm for nonnegative weights.

Contribution

It introduces a reduction of the problem to min-plus matrix multiplications and presents the first (1+ε)-approximation algorithm for nonnegative weights.

Findings

Reduction to O(n^2) min-plus matrix multiplications

First approximation algorithm with (1+ε) accuracy for nonnegative weights

Algorithm runs in near-optimal matrix multiplication time

Abstract

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 + {\epsilon})-approximation algorithm for the problem and the running time of our algorithm is \tilde(O)(n^\omega log^3(nW/{\epsilon}) / {\epsilon}), where O(n^\omega) is the time required for the classic n-by-n matrix multiplication and W is the maximum value of the weights.