Approximation Resistant Predicates from Pairwise Independence

Abstract.

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-CSP _q problem is UG-hard to approximate within ${\mathcal{O}}(kq^2)/q^k + \epsilon$.

• For the special case of q = 2, i.e., boolean variables, we can sharpen this bound to $(k +{\mathcal{O}}(k^{0.525}))/2^k +\epsilon$, improving upon the best previous bound of 2k/2^k + ∈ (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 ${\mathcal{O}}(k^{0.525})$ term can be replaced by the constant 4.