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arXiv · 2203.00671

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

Li Chen·Rasmus Kyng·Yang P. Liu·Richard Peng·Maximilian Probst Gutenberg·Sushant Sachdeva
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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 new dynamic graph data structure. Our framework extends to algorithms running in $m^{1+o(1)}$ time for computing flows that minimize general edge-separable convex functions to high accuracy. This gives almost-linear time algorithms for several problems including entropy-regularized optimal transport, matrix scaling, $p$-norm flows, and $p$-norm isotonic regression on arbitrary directed acyclic graphs.

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Li Chen, Rasmus Kyng, Yang P. Liu, Richard Peng, Maximilian Probst Gutenberg, Sushant Sachdeva. 2022-03-01. Maximum Flow and Minimum-Cost Flow in Almost-Linear Time. https://arxiv.org/abs/2203.00671

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