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DOI: 10.4230/LIPIcs.STACS.2018.11
URN: urn:nbn:de:0030-drops-85279
URL: https://drops.dagstuhl.de/opus/volltexte/2018/8527/
Berkholz, Christoph
The Relation between Polynomial Calculus, Sherali-Adams, and Sum-of-Squares Proofs
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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 andsum-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 | |
| Seminar: | 35th Symposium on Theoretical Aspects of Computer Science (STACS 2018) | |
| Issue date: | 2018 | |
| Date of publication: | 27.02.2018 |
