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

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.

Authors

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