Approximating Min-Mean-Cycle for Low-Diameter Graphs in Near-Optimal Time and Memory

Abstract

We revisit Min-Mean-Cycle, the classical problem of finding a cycle in a weighted directed graph with minimum mean weight. Despite an extensive algorithmic literature, previous work failed to achieve a near-linear runtime in the number of edges $m$. We propose an algorithm with near-linear runtime $\tilde{O}(m (W_{\max}/\epsilon)^2)$ for computing an $\epsilon$ additive approximation on graphs with polylogarithmic diameter and weights of magnitude at most $W_{\max}$. In particular, this is the first algorithm whose runtime scales in the number of vertices $n$ as $\tilde{O}(n^2)$ for the complete graph. Moreover---unconditionally on the diameter---the algorithm uses only $O(n)$ memory beyond reading the input, making it “memory-optimal". Our approach is based on solving a linear programming (LP) relaxation using entropic regularization, which reduces the LP to a Matrix Balancing problem---à la the popular reduction of Optimal Transport to Matrix Scaling. We then round the fractional LP solution using a variant of the classical Cycle-Canceling algorithm that is sped up to near-linear runtime at the expense of being approximate, and implemented in a memory-optimal manner. The algorithm is simple to implement and is competitive with the state-of-the-art methods in practice.