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Partially-Dynamic Maximum Flow in Dense Graphs

Authors Egor Kravchenko, Maximilian Probst Gutenberg

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LIPIcs.ICALP.2026.133.pdf
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Subject Classification

ACM Subject Classification
  • Theory of computation → Dynamic graph algorithms
Keywords
  • Maximum Flow
  • Dynamic Graph Algorithm
  • Data Structure

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Abstract

We give the first algorithms that, with high probability, maintain (1-ε)-approximate s-t maximum flow in an n-vertex undirected, capacitated graph undergoing either only edge insertions or only edge deletions in total update time Õ_ε(n²). For dense graphs, this yields polylogarithmic amortized update time, which was previously only obtained for the special case of uncapacitated graphs undergoing edge insertions. 
We develop the following two algorithms:  
- For graphs undergoing deletions, we generalize the congestion-balancing framework from [Aaron Bernstein et al., 2020], which was developed for maximum matching. We then show that this framework can be simulated on cut sparsifiers, which yields significant speed-ups. 
- For graphs undergoing insertions, we show that the sparsification techniques by Eppstein et al. [Eppstein et al., 1997] can be combined more directly with the techniques from Henzinger and Goranci [Goranci and Henzinger, 2023]. We thereby bypass the need to dynamize the more involved residual graph sparsification approach by Levin and Karger [Karger and Levine, 2015] suggested in [Goranci et al., 2025], and extend their result to capacitated graphs.

Cite As Get BibTex

Egor Kravchenko and Maximilian Probst Gutenberg. Partially-Dynamic Maximum Flow in Dense Graphs. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 133:1-133:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.ICALP.2026.133

Author Details

Egor Kravchenko
  • ETH Zurich, Switzerland
Maximilian Probst Gutenberg
  • ETH Zurich, Switzerland

Funding

  • Probst Gutenberg, Maximilian: The research leading to these results has received funding from grant no. 200021 204787 of the Swiss National Science Foundation.

Acknowledgements

The authors would like to thank Simon Meierhans for various insightful discussions.

References

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