The Complexity of Somewhat Approximation Resistant Predicates

A boolean predicate f : {0, 1} → {0, 1} is said to be somewhat approximation resistant if for some constant τ > |f −1(1)| 2k , given a τ -satisfiable instance of the MAX k-CSP(f) problem, it is NP-hard to find an assignment that strictly beats the naive algorithm that outputs a uniformly random assignment. Let τ(f) denote the supremum over all τ for which this holds. It is known that a predicate is somewhat approximation resistant precisely when its Fourier degree is at least 3. For such predicates, we give a characterization of the hardness gap (τ(f)− |f −1(1)| 2k ) up to a factor of O(k). We also give a similar characterization of the integrality gap for the natural SDP relaxation of MAX k-CSP(f) after Ω(n) rounds of the Lasserre hierarchy.