SizeDegree TradeOffs for SumsofSquares and Positivstellensatz Proofs
Abstract
We show that if a system of degree$k$ polynomial constraints on~$n$ Boolean variables has a SumsofSquares (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 sizedegree tradeoffs 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 NPhard 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 zerogap duality theorem for preordered vector spaces that admit an order unit, whose specialization to PS and SOS may be of independent interest.
 Publication:

arXiv eprints
 Pub Date:
 November 2018
 arXiv:
 arXiv:1811.01351
 Bibcode:
 2018arXiv181101351A
 Keywords:

 Computer Science  Computational Complexity;
 Computer Science  Logic in Computer Science