Theory of Computing ------------------- Title : Approximation Resistance on Satisfiable Instances for Sparse Predicates Authors : Sangxia Huang Volume : 10 Number : 14 Pages : 359-388 URL : https://theoryofcomputing.org/articles/v010a014 Abstract -------- For every integer $k \ge 3$, we prove that there is a predicate $P$ on $k$ Boolean variables with $2^{O~(k^{1/3})}$ accepting assignments that is approximation resistant even on _satisfiable_ instances. That is, given a _satisfiable_ CSP instance with constraint $P$, we cannot achieve better approximation ratio than simply picking random assignments. This improves the best previously known result by Hastad and Khot (Theory of Computing, 2005) who showed that a predicate on $k$ variables with $2^{O(k^{1/2})}$ accepting assignments is approximation resistant on satisfiable instances. Our construction is inspired by several recent developments. One is the idea of using direct sums to improve soundness of PCPs, developed by Siu On Chan (STOC, 2013). We also use techniques from Cenny Wenner (Theory of Computing, 2013) to construct PCPs with perfect completeness without relying on the $d$-to-1 Conjecture. A conference version of this paper appeared in the Proceedings of the 45th Annual ACM Symposium on Theory of Computing, 2013.