Let P:{0,1}k→{0,1} be a nontrivial k-ary predicate. Consider a
random instance of the constraint satisfaction problem CSP(P) on n
variables with Δn 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 \emph{refutation}
problem is to efficiently find a proof of unsatisfiability.
We show that whenever the predicate P supports a t-\emph{wise uniform}
probability distribution on its satisfying assignments, the sum of squares
(SOS) algorithm of degree d=Θ(Δ2/(t−1)logΔn)
(which runs in time nO(d)) \emph{cannot} refute a random instance of
CSP(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 et al. [LRS15], our result also
implies that \emph{any} polynomial-size semidefinite programming relaxation for
refutation requires at least Ω(n(t+1)/2) constraints.
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 et al. [AOW15] and Raghavendra et al. [RRS16], this full three-way
tradeoff is \emph{tight}, up to lower-order factors.Comment: 39 pages, 1 figur