Size-Degree Trade-Offs for Sums-of-Squares and Positivstellensatz Proofs

Albert Atserias, Tuomas Hakoniemi, Michael Wagner
2019 Computational Complexity Conference  
We show that if a system of degree-k polynomial constraints on n Boolean variables has a Sums-of-Squares (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 size-degree trade-offs for SOS and PS that match their analogues for weaker proof systems such as Resolution, Polynomial Calculus, and the proof
more » ... tems 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 NP-hard 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 zero-gap duality theorem for pre-ordered vector spaces that admit an order unit, whose specialization to PS and SOS may be of independent interest.
doi:10.4230/lipics.ccc.2019.24 dblp:conf/coco/AtseriasH19 fatcat:ega7hny7tngcbbl342szz7f3ye