Maximum Flow and Minimum-Cost Flow in Almost-Linear Time

by Li Chen, Rasmus Kyng, Yang P. Liu, Richard Peng, Maximilian Probst Gutenberg, and Sushant Sachdeva

Oded's comments

I guess (since I don't know) that this paper was featured on all BLOGs a couple of weeks ago, and it definitely does not need my recommendation; still, it feels odd to ignore it. Another recommendation, which probably bursts to an open door, is watching a talk on this work; Part 1 (approximately 20 minutes) provides a good overview of the context and contents of this work.

The original abstract

We give an algorithm that computes exact maximum flows and minimum-cost flows on directed graphs with m edges and polynomially bounded integral demands, costs, and capacities in $m^{1+o(1)}$ time. Our algorithm builds the flow through a sequence of $m^{1+o(1)}$ approximate undirected minimum-ratio cycles, each of which is computed and processed in amortized $m^{o(1)}$ time using a dynamic data structure.

Our framework extends to an algorithm running in $m^{1+o(1)}$ time for computing flows that minimize general edge-separable convex functions to high accuracy. This gives an almost-linear time algorithm for several problems including entropy-regularized optimal transport, matrix scaling, p-norm flows, and isotonic regression.

Available from ArXiv 2203.00671.

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