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Sum of Squares Bounds for the Ordering Principle

Author Aaron Potechin

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Author Details

Aaron Potechin
  • University of Chicago, IL, USA

Funding

This work was supported by the Knut and Alice Wallenberg Foundation, the European Research Council, and the Swedish Research Council.

Acknowledgements

The author would like to thank Marc Vinyals for proposing this problem. The author would also like to thank Prahladh Harsha, Toniann Pitassi, and anonymous reviewers for helpful comments on this paper.

Cite AsGet BibTex

Aaron Potechin. Sum of Squares Bounds for the Ordering Principle. In 35th Computational Complexity Conference (CCC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 169, pp. 38:1-38:37, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
https://doi.org/10.4230/LIPIcs.CCC.2020.38

Abstract

In this paper, we analyze the sum of squares hierarchy (SOS) on the ordering principle on n elements (which has N = Θ(n²) variables). We prove that degree O(√nlog(n)) SOS can prove the ordering principle. We then show that this upper bound is essentially tight by proving that for any ε > 0, SOS requires degree Ω(n^(1/2 - ε)) to prove the ordering principle.

Subject Classification

ACM Subject Classification
  • Theory of computation → Proof complexity
Keywords
  • sum of squares hierarchy
  • proof complexity
  • ordering principle

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