The complexity of somewhat approximation resistant predicates

A boolean predicate f:{0,1}^k → {0,1} is said to be somewhat approximation resistant if for some constant τ > |f^-1(1)|/ 2^k, 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)|/2^k) up to a factor of O(k⁵). We show that the hardness gap is determined by two factors: - The nearest Hamming distance of f to a function g of Fourier degree at most 2, which is related to the Fourier mass of f on coefficients of degree 3 or higher. - Whether f is monotonically below g. When the Hamming distance is small and f is monotonically below g, we give an SDP-based approximation algorithm and hardness results otherwise. 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.