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-in fact, there is a natural barrier which precludes such a runtime for solving Min-Mean-Cycle exactly. Here, we give a much faster approximation algorithm that, for graphs with polylogarithmic diameter, has near-linear runtime. In particular, this is the first algorithm whose runtime for the complete graph scales in the number of vertices n as Õ(n 2 ). Moreover-unconditionally on the diameter-the algorithm uses only O(n) memory beyond reading the input, making it "memory-optimal". The algorithm is also simple to implement and has remarkable practical performance.Our approach is based on solving a linear programming (LP) relaxation using entropic regularization, which effectively reduces the LP to a Matrix Balancing problem-á la the popular reduction of Optimal Transport to Matrix Scaling. We then round the fractional LP solution using a variant of the classical Cycle-Cancelling algorithm that is sped up to near-linear runtime at the expense of being approximate, and implemented in a memory-optimal manner.We also provide an alternative algorithm with slightly faster theoretical runtime, albeit worse memory usage and practicality. This algorithm uses the same rounding procedure, but solves the LP relaxation by leveraging recent developments in area-convexity regularization. Its runtime scales inversely in the approximation accuracy, which we show is optimal-barring a major breakthrough in algorithmic graph theory, namely faster Shortest Paths algorithms.

Asymptotics for semi-discrete entropic optimal transport

“…Cuturi introduced this work to the machine learning community in 2013 [19], which led to an explosion of interest in optimal transport for applications [47]. Subsequently, the entropic penalty has been applied to variants of the optimal transport problem, where it also leads to fast and practical algorithms [2,6,7,12].…”

Quantum algorithms for matrix scaling and matrix balancing

“…In mathematics, it has been used as a common tool in practical linear algebra computations [LG04,Bra10,PC11,OCPB16], but also in statistics [Sin64], optimization [RS89], and for strengthening the Sylvester-Gallai theorem [BDYW11]. Matrix balancing has a similarly wide variety of applications, including pre-conditioning to make practical matrix computations more stable (as mentioned above), and approximating the min-mean-cycle in a weighted graph [AP20a]. Many more applications of matrix scaling and balancing are mentioned in [LSW00,Ide16,GO18].…”