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

April, 2022

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 m1+o(1) time. Our algorithm builds the flow through a sequence of m1+o(1) approximate undirected minimum-ratio cycles, each of which is computed and processed in amortized mo(1) time using a new dynamic graph data structure.

Our framework extends to algorithms running in m1+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.