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Expander Decomposition with Almost Optimal Overhead

Authors Nikhil Bansal , Arun Jambulapati, Thatchaphol Saranurak

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Subject Classification

ACM Subject Classification
  • Theory of computation → Graph algorithms analysis
  • Theory of computation → Approximation algorithms analysis
Keywords
  • Graph algorithms
  • expander decomposition
  • flow expansion
  • sparse cuts

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Abstract

We present the first polynomial-time algorithm for computing a near-optimal flow-expander decomposition. Given a graph G and a parameter ϕ, our algorithm removes at most a ϕlog^{1+o(1)}n fraction of edges so that every remaining connected component is a ϕ-flow-expander (a stronger guarantee than being a ϕ-cut-expander). This achieves overhead log^{1+o(1)}n, nearly matching the Ω(log n) graph-theoretic lower bound that already holds for cut-expander decompositions, up to a log^{o(1)}n factor. Prior polynomial-time algorithms required removing O(ϕlog^{1.5}n) and O(ϕlog²n) fractions of edges to guarantee ϕ-cut-expander and ϕ-flow-expander components, respectively.

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Nikhil Bansal, Arun Jambulapati, and Thatchaphol Saranurak. Expander Decomposition with Almost Optimal Overhead. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 22:1-22:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.ICALP.2026.22

Author Details

Nikhil Bansal
  • University of Michigan, Ann Arbor, MI, USA
Arun Jambulapati
  • Independent Researcher, USA
Thatchaphol Saranurak
  • University of Michigan, Ann Arbor, MI, USA

Funding

  • Bansal, Nikhil: Supported by NSF awards CCF-2327011 and CCF-2504995.
  • Jambulapati, Arun: Supported by the NWO VICI grant 639.023.812 awarded to NB.
  • Saranurak, Thatchaphol: Supported by NSF Grant CCF-2238138 and a Sloan Fellowship.

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