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An Almost-Linear Time Algorithm for Maximum Flow and More (Invited Talk)

Author Rasmus Kyng

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Rasmus Kyng
  • ETH Zürich, Switzerland

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Rasmus Kyng. An Almost-Linear Time Algorithm for Maximum Flow and More (Invited Talk). In 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 261, p. 2:1, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)


In this talk, I will explain a new algorithm for computing exact maximum and minimum-cost flows in almost-linear time, settling the time complexity of these basic graph problems up to subpolynomial factors. Our algorithm uses a novel interior point method that builds the optimal flow as a sequence of approximate minimum-ratio cycles, each of which is computed and processed very efficiently using a new dynamic data structure. By well-known reductions, our result implies almost-linear time algorithms for several problems including bipartite matching, optimal transport, and undirected vertex connectivity. Our framework also extends to minimizing general edge-separable convex functions to high accuracy, yielding the first almost-linear time algorithms for many other problems including entropy-regularized optimal transport, matrix scaling, p-norm flows, and isotonic regression. This talk is based on joint work with Li Chen, Yang P. Liu, Richard Peng, Maximilian Probst Gutenberg, and Sushant Sachdeva [Chen et al., 2022]. Our result appeared in FOCS'22 and won the FOCS best paper award.

Subject Classification

ACM Subject Classification
  • Theory of computation → Network flows
  • Theory of computation → Sparsification and spanners
  • Theory of computation → Dynamic graph algorithms
  • Maximum flow
  • Minimum cost flow
  • Data structures
  • Interior point methods
  • Convex optimization


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  1. Li Chen, Rasmus Kyng, Yang P. Liu, Richard Peng, Maximilian Probst Gutenberg, and Sushant Sachdeva. Maximum flow and minimum-cost flow in almost-linear time. In 2022 IEEE 63rd Annual Symposium on Foundations of Computer Science (FOCS), pages 612-623, 2022. URL:
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