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

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 nΩ(d) for values of d=d(n) from constant all the way up to nδ for some universal constantδ. This shows that the nO(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 [Kraj́íček '04] and [Dantchev and Riis'03], and then applying a restriction argument as in [Atserias, Müller, and Oliva '13] and [Atserias, Lauria, and Nordström '14]. This yields a generic method of amplifying SOS degree lower bounds to size lower bounds, and also generalizes the approach in [ALN14] to obtain size lower bounds for the proof systems resolution, polynomial calculus, and Sherali-Adams from lower bounds on width, degree, and rank, respectively.