APPROXIMATION RESISTANT PREDICATES FROM PAIRWISE INDEPENDENCE

We study the approximability of predicates on k variables from a domain [q], and give a new sufficient condition for such predicates to be approximation resistant under the Unique Games Conjecture. Specifically, we show that a predicate P is approximation resistant if there exists a balanced pairwise independent distribution over [q](k) whose support is contained in the set of satisfying assignments to P. Using constructions of pairwise independent distributions this result implies that For general k >= 3 and q <= 2, the Max k-CSPq problem is UG-hard to approximate within O(kq(2))/q(k) + epsilon. For the special case of q = 2, i.e., boolean variables, we can sharpen this bound to (k + O(k(0.525)))/2(k) + epsilon, improving upon the best previous bound of 2k/2(k) + epsilon (Samorodnitsky and Trevisan, STOC'06) by essentially a factor 2. Finally, again for q = 2, assuming that the famous Hadamard Conjecture is true, this can be improved even further, and the O(k(0.525)) term can be replaced by the constant 4.