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Multiplicative Error Set System Sparsification: A Simpler Proof via Chain Length Contraction

Authors Joshua Brakensiek , Venkatesan Guruswami , Aaron Putterman

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ACM Subject Classification
  • Theory of computation → Constraint and logic programming
Keywords
  • constraint satisfaction problem
  • chain length
  • sparsification
  • VC dimension

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Abstract

The chain length of a set family 𝒮 ⊆ 2^[m] is the largest ascending sequence of sets in containment order in the union-closure of S. In this work, we provide a significantly simpler and more optimal characterization of the sparsifiability of set systems in terms of their chain length, improving on the work of Brakensiek and Guruswami [STOC 2025]. Our proof relies on a generalization of Karger’s [SODA 1993] famous contraction algorithm and its recent linear algebraic extensions [Khanna-Putterman-Sudan SODA 2024], and our resulting bounds show that, just as VC dimension characterizes the additive sparsifiability of a set system, chain length governs the multiplicative sparsifiability. As a corollary, we obtain improved bounds for weighted CSP sparsification.

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Joshua Brakensiek, Venkatesan Guruswami, and Aaron Putterman. Multiplicative Error Set System Sparsification: A Simpler Proof via Chain Length Contraction. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 44:1-44:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.ICALP.2026.44

Author Details

Joshua Brakensiek
  • Department of Electrical Engineering and Computer Sciences, University of California, Berkeley, CA, USA
Venkatesan Guruswami
  • Simons Institute for the Theory of Computing and the University of California, Berkeley, CA, USA
Aaron Putterman
  • School of Engineering and Applied Sciences, Harvard University, Cambridge, MA, USA

Funding

  • Brakensiek, Joshua: Supported in part by the Simons Investigator award of Venkatesan Guruswami and NSF grants CCF-2211972 and DMS-2503280.
  • Guruswami, Venkatesan: Supported in part by a Simons Investigator award and NSF award CCF-2211972.
  • Putterman, Aaron: Supported in part by the Simons Investigator Awards of Madhu Sudan and Salil Vadhan and AFOSR award FA9550-25-1-0112.

Acknowledgements

We thank anonymous reviewers for many helpful comments improving the presentation of this paper. We also thank an anonymous reviewer of [Joshua Brakensiek and Venkatesan Guruswami, 2025] whose comments partially inspired Theorem 22. A. P. thanks Sanjeev Khanna and Madhu Sudan for helpful conversations.

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