Size-Degree Trade-Offs for Sums-of-Squares and Positivstellensatz Proofs

Authors Albert Atserias , Tuomas Hakoniemi

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Albert Atserias
  • Universitat Politècnica de Catalunya, Barcelona, Spain
Tuomas Hakoniemi
  • Universitat Politècnica de Catalunya, Barcelona, Spain


We are grateful to Michal Garlik, Moritz Müller and Aaron Potechin for comments on an earlier version of this paper. We are also grateful to Jakob Nordström for initiating a discussion on the several variants of the definition of monomial size as discussed in Section 2.

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Albert Atserias and Tuomas Hakoniemi. Size-Degree Trade-Offs for Sums-of-Squares and Positivstellensatz Proofs. In 34th Computational Complexity Conference (CCC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 137, pp. 24:1-24:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)


We show that if a system of degree-k polynomial constraints on n Boolean variables has a Sums-of-Squares (SOS) proof of unsatisfiability with at most s many monomials, then it also has one whose degree is of the order of the square root of n log s plus k. A similar statement holds for the more general Positivstellensatz (PS) proofs. This establishes size-degree trade-offs for SOS and PS that match their analogues for weaker proof systems such as Resolution, Polynomial Calculus, and the proof systems for the LP and SDP hierarchies of Lovász and Schrijver. As a corollary to this, and to the known degree lower bounds, we get optimal integrality gaps for exponential size SOS proofs for sparse random instances of the standard NP-hard constraint optimization problems. We also get exponential size SOS lower bounds for Tseitin and Knapsack formulas. The proof of our main result relies on a zero-gap duality theorem for pre-ordered vector spaces that admit an order unit, whose specialization to PS and SOS may be of independent interest.

Subject Classification

ACM Subject Classification
  • Theory of computation → Proof complexity
  • Proof complexity
  • semialgebraic proof systems
  • Sums-of-Squares
  • Positivstellensatz
  • trade-offs
  • lower bounds
  • monomial size
  • degree


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