Sum of squares lower bounds for refuting any CSP

Abstract

Let P:{0,1}k → {0,1} be a nontrivial k-ary predicate. Consider a random instance of the constraint satisfaction problem (P) on n variables with Δ n constraints, each being P applied to k randomly chosen literals. Provided the constraint density satisfies Δ ≫ 1, such an instance is unsatisfiable with high probability. The refutation problem is to efficiently find a proof of unsatisfiability. We show that whenever the predicate P supports a t-wise uniform probability distribution on its satisfying assignments, the sum of squares (SOS) algorithm of degree d = Θ(n/Δ2/(t-1) logΔ) (which runs in time nO(d)) cannot refute a random instance of (P). In particular, the polynomial-time SOS algorithm requires Ω(n(t+1)/2) constraints to refute random instances of CSP(P) when P supports a t-wise uniform distribution on its satisfying assignments. Together with recent work of Lee, Raghavendra, Steurer (2015), our result also implies that any polynomial-size semidefinite programming relaxation for refutation requires at least Ω(n(t+1)/2) constraints. More generally, we consider the δ-refutation problem, in which the goal is to certify that at most a (1 - δ)-fraction of constraints can be simultaneously satisfied. We show that if P is δ-close to supporting a t-wise uniform distribution on satisfying assignments, then the degree-Θ(n/Δ2/(t - 1) logΔ) SOS algorithm cannot (δ+o(1))-refute a random instance of CSP(P). This is the first result to show a distinction between the degree SOS needs to solve the refutation problem and the degree it needs to solve the harder δ-refutation problem. Our results (which also extend with no change to CSPs over larger alphabets) subsume all previously known lower bounds for semialgebraic refutation of random CSPs. For every constraint predicate P, they give a three-way hardness tradeoff between the density of constraints, the SOS degree (hence running time), and the strength of the refutation. By recent algorithmic results of Allen, O'Donnell, Witmer (2015) and Raghavendra, Rao, Schramm (2016), this full three-way tradeoff is tight, up to lower-order factors.