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

Li Chen; Rasmus Kyng; Yang P. Liu; Richard Peng; Maximilian Probst Gutenberg; Sushant Sachdeva

Paper

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

Abstract

We give an algorithm that computes exact maximum flows and minimum-cost flows on directed graphs with

edges and polynomially bounded integral demands, costs, and capacities in

time. Our algorithm builds the flow through a sequence of

approximate undirected minimum-ratio cycles, each of which is computed and processed in amortized

time using a new dynamic graph data structure. Our framework extends to algorithms running in

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,

-norm flows, and

-norm isotonic regression on arbitrary directed acyclic graphs.