Circumventing d-to-1 for Approximation Resistance of Satisfiable Predicates Strictly Containing Parity of Width at Least Four

Håstad established that any predicate P ⊆ {0,1}m containing Parity of width at least three is approximation resistant for almost-satisfiable instances. In comparison to for example the approximation hardness of 3SAT, this general result however left open the hardness of perfectly-satisfiable instances. This limitation was addressed by O’Donnell and Wu, and subsequently generalized by Huang, to show the threshold result that predicates strictly containing Parity of width at least three are approximation resistant also for perfectlysatisfiable instances, assuming the d-to-1 Conjecture. We extend modern hardness-of-approximation techniques by Mossel et al., eliminating the dependency on projection degrees for a special case of decoupling/invariance and—when ∗A preliminary version of this paper appeared in the International Workshop on Approximation Algorithms for Combinatorial Optimization Problems (APPROX’12) [28], and a technical preprint appeared in the Electronic Colloquium on Computational Complexity (ECCC’12), 2012 [27]. †Supported by ERC Advanced Investigator Grant 226203. ACM Classification: G.1.2., G.1.6, G.3 AMS Classification: 68Q17, 68Q25, 68Q87, 68W25, 90C59