Abstract
For every integer $k \ge 3$, we prove that there is a predicate $P$ on $k$ Boolean variables with $2^{\widetilde{O}(k^{1/3})}$ 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 Håstad 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.
Highlights
In this article we study optimization of constraint satisfaction problems with k variables in each constraint (MAX-k-CSP)
A k-CSP instance contains a set of Boolean variables and constraints, where each constraint is expressed by a Boolean predicate on k literals
Given a predicate P : {−1, 1}k → {0, 1} on k Boolean inputs, we can consider the MAX-P problem, a special case of MAX-k-CSP where all constraints are expressed by the same Boolean predicate P applied to literals of k distinct variables
Summary
In this article we study optimization of constraint satisfaction problems with k variables in each constraint (MAX-k-CSP). There have been only a handful of results: Håstad [14] proved that k-SAT is approximation resistant for satisfiable instances The sparsest such predicate known on k variables has 2O(k1/2) accepting assignments, given by Håstad and Khot [16]. From the PCP perspective, this requires PCPs that always accept correct proofs of correct statements Is this a natural property to have, given the challenge of getting proofs with perfect completeness as discussed above, understanding approximability of k-CSP on satisfiable instances may lead to new tools in both algorithms and hardness results. There is a predicate of arity K with 2O(K1/3) accepting assignments that is approximation resistant on satisfiable instances This improves the best previous known result of 2O(K1/2) of Håstad and Khot [16]. Similar techniques were developed in other works such as [25] as well as in [5]
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