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The Relation between Polynomial Calculus, Sherali-Adams, and Sum-of-Squares Proofs

Author Christoph Berkholz

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Christoph Berkholz

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Christoph Berkholz. The Relation between Polynomial Calculus, Sherali-Adams, and Sum-of-Squares Proofs. In 35th Symposium on Theoretical Aspects of Computer Science (STACS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 96, pp. 11:1-11:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018) https://doi.org/10.4230/LIPIcs.STACS.2018.11

Abstract

We relate different approaches for proving the unsatisfiability of a system of real polynomial equations over Boolean variables. On the one hand, there are the static proof systems Sherali-Adams and
sum-of-squares (a.k.a. Lasserre), which are based on linear and
semi-definite programming relaxations. On the other hand, we
consider polynomial calculus, which is a dynamic algebraic proof
system that models Gröbner basis computations.
  
Our first result is that sum-of-squares simulates polynomial
calculus: any polynomial calculus refutation of degree d can be
transformed into a sum-of-squares refutation of degree 2d and only
polynomial increase in size.
In contrast, our second result shows that this is not the case for Sherali-Adams: there are systems of polynomial equations that have polynomial calculus refutations of degree 3 and polynomial size, but require Sherali-Adams refutations of large degree and exponential size.

A corollary of our first result is that the proof systems
Positivstellensatz and Positivstellensatz Calculus, which have been separated over non-Boolean polynomials, simulate
each other in the presence of Boolean axioms.

Subject Classification

Keywords
  • Proof Complexity
  • Polynomial Calculus
  • Sum-of-Squares
  • Sherali-Adams

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