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Average-Case Hardness of Binary-Encoded Clique in Proof and Communication Complexity

Authors Susanna F. de Rezende , David Engström , Yassine Ghannane , Duri Andrea Janett , Artur Riazanov

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LIPIcs.ICALP.2026.151.pdf
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Subject Classification

ACM Subject Classification
  • Theory of computation → Computational complexity and cryptography
Keywords
  • proof complexity
  • communication complexity
  • cutting planes
  • bounded-depth resolution over parities
  • clique problem
  • average-case hardness
  • binary encoding

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Abstract

We study the average-case hardness of establishing that a graph does not have a large clique in both proof and communication complexity. We show exponential lower bounds on the length of cutting planes and bounded-depth resolution over parities refutations of the binary encoding of clique formulas on randomly sampled dense graphs. Moreover, we show that the randomized communication complexity of finding a falsified clause in these formulas is polynomial.

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Susanna F. de Rezende, David Engström, Yassine Ghannane, Duri Andrea Janett, and Artur Riazanov. Average-Case Hardness of Binary-Encoded Clique in Proof and Communication Complexity. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 151:1-151:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.ICALP.2026.151

Author Details

Susanna F. de Rezende
  • Lund University, Sweden
David Engström
  • Lund University, Sweden
Yassine Ghannane
  • University of Copenhagen, Denmark
  • Lund University, Sweden
Duri Andrea Janett
  • University of Copenhagen, Denmark
  • Lund University, Sweden
Artur Riazanov
  • EPFL, Lausanne, Switzerland

Funding

  • de Rezende, Susanna F.: Received funding from the Knut and Alice Wallenberg grant KAW 2023.0116, ELLIIT, and the Swedish Research Council grant 2021-05104.
  • Engström, David: Received funding from the Knut and Alice Wallenberg grant KAW 2023.0116, ELLIIT, and the Swedish Research Council grant 2021-05104.
  • Ghannane, Yassine: Supported by the Independent Research Fund Denmark grant 9040-00389B.
  • Janett, Duri Andrea: Supported by the Independent Research Fund Denmark grant 9040-00389B.
  • Riazanov, Artur: Supported by Swiss State Secretariat for Education, Research, and Innovation (SERI) under contract number MB22.00026.

Acknowledgements

We would like to thank Dmitry Itsykson and Kilian Risse for insightful discussions. We also gratefully acknowledge that we have benefited greatly from being part of Basic Algorithms Research Copenhagen (BARC) environment financed by the Villum Investigator grant 54451.

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