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 falls short of a near-linear runtime in the number of edges m. We propose an approximation algorithm that, for graphs with polylogarithmic diameter, achieves a near-linear runtime. In particular, this is the first algorithm whose runtime scales in the number of vertices n as O(n2) 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 relaxation using entropic regularization, which reduces the problem to Matrix Balancing -- \'a la the popular reduction of Optimal Transport to Matrix Scaling. The algorithm is practical and simple to implement.