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Length-Constrained Directed Expander Decomposition and Length-Constrained Vertex-Capacitated Flow Shortcuts

Authors Bernhard Haeupler , Yaowei Long , Thatchaphol Saranurak , Shengzhe Wang

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  • Theory of computation → Graph algorithms analysis
Keywords
  • Length-Constrained Expander
  • Expander Decomposition
  • Shortcut

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Abstract

We show the existence of length-constrained expander decomposition in directed graphs and undirected vertex-capacitated graphs. Previously, its existence was shown only in undirected edge-capacitated graphs [Bernhard Haeupler et al., 2022; Haeupler et al., 2024]. Along the way, we prove the multi-commodity maxflow-mincut theorems for length-constrained expansion in both directed and undirected vertex-capacitated graphs. Based on our decomposition, we build a length-constrained flow shortcut for undirected vertex-capacitated graphs, which roughly speaking is a set of edges and vertices added to the graph so that every multi-commodity flow demand can be routed with approximately the same vertex-congestion and length, but all flow paths only contain few edges. This generalizes the shortcut for undirected edge-capacitated graphs from [Bernhard Haeupler et al., 2024]. Length-constrained expander decomposition and flow shortcuts have been crucial in the recent algorithms in undirected edge-capacitated graphs [Bernhard Haeupler et al., 2024; Haeupler et al., 2024]. Our work thus serves as a foundation to generalize these concepts to directed and vertex-capacitated graphs.

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Bernhard Haeupler, Yaowei Long, Thatchaphol Saranurak, and Shengzhe Wang. Length-Constrained Directed Expander Decomposition and Length-Constrained Vertex-Capacitated Flow Shortcuts. In 33rd Annual European Symposium on Algorithms (ESA 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 351, pp. 107:1-107:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.ESA.2025.107

Author Details

Bernhard Haeupler
  • INSAIT, Sofia University "St. Kliment Ohridski", Bulgaria
  • ETH Zürich, Switzerland
Yaowei Long
  • University of Michigan, Ann Arbor, MI, USA
Thatchaphol Saranurak
  • University of Michigan, Ann Arbor, MI, USA
Shengzhe Wang
  • ETH Zürich, Switzerland

Funding

Part of this work was done while at INSAIT, Sofia University "St. Kliment Ohridski", Bulgaria. This work was partially funded from the Ministry of Education and Science of Bulgaria (support for INSAIT, part of the Bulgarian National Roadmap for Research Infrastructure).
  • Haeupler, Bernhard: Supported by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program (ERC grant agreement 949272).
  • Saranurak, Thatchaphol: Supported by NSF Grant CCF-2238138.
  • Wang, Shengzhe: Supported by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (ERC grant agreement No 949272 and No 101171685).

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