A boolean predicate f:{0,1}k → {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(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. © 2014 Springer-Verlag.
CITATION STYLE
Khot, S., Tulsiani, M., & Worah, P. (2014). The complexity of somewhat approximation resistant predicates. In Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) (Vol. 8572 LNCS, pp. 689–700). Springer Verlag. https://doi.org/10.1007/978-3-662-43948-7_57
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