License: Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.STACS.2018.11
URN: urn:nbn:de:0030-drops-85279
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2018/8527/
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Berkholz, Christoph

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

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LIPIcs-STACS-2018-11.pdf (0.5 MB)

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.

BibTeX - Entry

@InProceedings{berkholz:LIPIcs:2018:8527,
  author =	{Christoph Berkholz},
  title =	{{The Relation between Polynomial Calculus, Sherali-Adams, and Sum-of-Squares Proofs}},
  booktitle =	{35th Symposium on Theoretical Aspects of Computer Science (STACS 2018)},
  pages =	{11:1--11:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-062-0},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{96},
  editor =	{Rolf Niedermeier and Brigitte Vall{\'e}e},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2018/8527},
  URN =		{urn:nbn:de:0030-drops-85279},
  doi =		{10.4230/LIPIcs.STACS.2018.11},
  annote =	{Keywords: Proof Complexity, Polynomial Calculus, Sum-of-Squares, Sherali-Adams}
}

Keywords: Proof Complexity, Polynomial Calculus, Sum-of-Squares, Sherali-Adams
Collection: 35th Symposium on Theoretical Aspects of Computer Science (STACS 2018)
Issue Date: 2018
Date of publication: 27.02.2018

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