Tight Size-Degree Bounds for Sums-of-Squares Proofs

Abstract

We exhibit families of 4-CNF formulas over n variables that have sums-of-squares (SOS) proofs of unsatisfiability of degree (a.k.a. rank) d but require SOS proofs of size ${n^{\Omega{(d)}}}$ for values of d = d(n) from constant all the way up to ${n^{\delta}}$ for some universal constant ${\delta}$. This shows that the ${{n^{{\rm O}{(d)}}}}$ running time obtained by using the Lasserre semidefinite programming relaxations to find degree-d SOS proofs is optimal up to constant factors in the exponent. We establish this result by combining NP-reductions expressible as low-degree SOS derivations with the idea of relativizing CNF formulas in Krajíček (Arch Math Log 43(4):427–441, 2004) and Dantchev & Riis (Proceedings of the 17th international workshop on computer science logic (CSL ’03), 2003) and then applying a restriction argument as in Atserias et al. (J Symb Log 80(2):450–476, 2015; ACM Trans Comput Log 17:19:1–19:30, 2016). This yields a generic method of amplifying SOS degree lower bounds to size lower bounds and also generalizes the approach used in Atserias et al. (2016) to obtain size lower bounds for the proof systems resolution, polynomial calculus, and Sherali–Adams from lower bounds on width, degree, and rank, respectively.