Trade-Offs Between Size and Degree in Polynomial Calculus

Authors Guillaume Lagarde, Jakob Nordström , Dmitry Sokolov, Joseph Swernofsky

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Author Details

Guillaume Lagarde
  • LaBRI, Bordeaux, France
Jakob Nordström
  • University of Copenhagen, Denmark
  • KTH Royal Institute of Technology, Stockholm, Sweden
Dmitry Sokolov
  • Lund University, Sweden
  • University of Copenhagen, Denmark
Joseph Swernofsky
  • KTH Royal Institute of Technology, Stockholm, Sweden


The first, second, and fourth authors were funded by the Knut and Alice Wallenberg grant KAW 2016.0066 and the third author by the Knut and Alice Wallenberg grant KAW 2016.0433. In addition, the second author was supported by the Swedish Research Council grants 621-2012-5645 and 2016-00782.

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Guillaume Lagarde, Jakob Nordström, Dmitry Sokolov, and Joseph Swernofsky. Trade-Offs Between Size and Degree in Polynomial Calculus. In 11th Innovations in Theoretical Computer Science Conference (ITCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 151, pp. 72:1-72:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)


Building on [Clegg et al. '96], [Impagliazzo et al. '99] established that if an unsatisfiable k-CNF formula over n variables has a refutation of size S in the polynomial calculus resolution proof system, then this formula also has a refutation of degree k + O(√(n log S)). The proof of this works by converting a small-size refutation into a small-degree one, but at the expense of increasing the proof size exponentially. This raises the question of whether it is possible to achieve both small size and small degree in the same refutation, or whether the exponential blow-up is inherent. Using and extending ideas from [Thapen '16], who studied the analogous question for the resolution proof system, we prove that a strong size-degree trade-off is necessary.

Subject Classification

ACM Subject Classification
  • Theory of computation → Proof complexity
  • proof complexity
  • polynomial calculus
  • polynomial calculus resolution
  • PCR
  • size-degree trade-off
  • resolution
  • colored polynomial local search


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