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

2025/04/09 by Li Chen, Rasmus Kyng, Yang P. Liu +4
Computer Science · #Complexity and Algorithms in Graphs #Optimization and Search Problems #Computational Geometry and Mesh Generation

paper · doi:10.1145/3728631

Abstract

We present 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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