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arXiv · 2004.03114

Approximating Min-Mean-Cycle for low-diameter graphs in near-optimal time and memory

Jason M. Altschuler·Pablo A. Parrilo
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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 $\tilde{O}(n^2)$ 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 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.

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Jason M. Altschuler, Pablo A. Parrilo. 2020-04-07. Approximating Min-Mean-Cycle for low-diameter graphs in near-optimal time and memory. https://doi.org/10.1137/21m1439390

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