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Near-Optimal Algorithm for Directed Expander Decompositions

Authors Aurelio L. Sulser, Maximilian Probst Gutenberg

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ACM Subject Classification
  • Theory of computation → Dynamic graph algorithms
Keywords
  • Directed Expander Decomposition
  • Push-Pull-Relabel Algorithm

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Abstract

In this work, we present the first algorithm to compute expander decompositions in an m-edge directed graph with near-optimal time Õ(m). Further, our algorithm can maintain such a decomposition in a dynamic graph and again obtains near-optimal update times. Our result improves over previous algorithms [Bernstein et al., 2020; Hua et al., 2023] that only obtained algorithms optimal up to subpolynomial factors.
In order to obtain our new algorithm, we present a new push-pull-relabel flow framework that generalizes the classic push-relabel flow algorithm [Goldberg and Tarjan, 1988] which was later dynamized for computing expander decompositions in undirected graphs [Henzinger et al., 2020; Saranurak and Wang, 2019]. We then show that the flow problems formulated in recent work [Hua et al., 2023] to decompose directed graphs can be solved much more efficiently in the push-pull-relabel flow framework. 
Recently, our algorithm has already been employed to obtain the currently fastest algorithm to compute min-cost flows [Van Den Brand et al., 2024]. We further believe that our algorithm can be used to speed-up and simplify recent breakthroughs in combinatorial graph algorithms towards fast maximum flow algorithms [Chuzhoy and Khanna, 2024; Chuzhoy and Khanna, 2024; Bernstein et al., 2024].

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Aurelio L. Sulser and Maximilian Probst Gutenberg. Near-Optimal Algorithm for Directed Expander Decompositions. In 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 334, pp. 132:1-132:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.ICALP.2025.132

Author Details

Aurelio L. Sulser
  • ETH Zurich, Switzerland
Maximilian Probst Gutenberg
  • ETH Zurich, Switzerland

Funding

The research leading to these results has received funding from the starting grant "A New Paradigm for Flow and Cut Algorithms" (no. TMSGI2_218022) of the Swiss National Science Foundation.

Acknowledgements

The authors are thankful for many fruitful discussions and feedback on an earlier version of the paper from Rasmus Kyng, Simon Meierhans and Thatchaphol Saranurak.

References

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