Paper Info
Title
The Relation between Polynomial Calculus, Sherali-Adams, and Sum-of-Squares Proofs.
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 Grebner 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 degree Omega(root n/log n) 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.
Year
DOI
Venue
2018
10.4230/LIPIcs.STACS.2018.11
Leibniz International Proceedings in Informatics
Keywords
DocType
Volume
Proof Complexity,Polynomial Calculus,Sum-of-Squares,Sherali-Adams
Conference
96
ISSN
Citations 
PageRank 
1868-8969
2
0.37
References 
Authors
0
1
Name
Order
Citations
PageRank
Christoph Berkholz1497.03