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

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

- Archive id: doi:10.1145/3728631
- Published: 2025-04-09
- Venue: Journal of the ACM
- DOI: https://doi.org/10.1145/3728631
- Citations: 22
- Source URL: https://doi.org/10.1145/3728631
- PDF: https://dl.acm.org/doi/pdf/10.1145/3728631
- Provenance: openalex

## 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.

_Abstract reproduced under CC-BY (publisher license, via OpenAlex)._

Topics: [Algorithms & Complexity Breakthroughs](https://space-kitty.com/topics/algorithms-complexity)

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