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 UGhard to approximate within O(kq 2 )/q k + . • For the special case of q = 2, i.e., boolean variables, we can sharpen this bound to (k + O(k 0.525 ))/2 k + , 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 O(k 0.525 ) term can be replaced by the constant 4.

Abstract-We study the inapproximability of Vertex Cover and Independent Set on degree d graphs. We prove that:• Vertex Cover is Unique Games-hard to approximate to within a factor 2 − (2

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Recently Raghavendra and Tan (SODA 2012) gave a 0.85-approximation algorithm for the Max Bisection problem. We improve their algorithm to a 0.8776-approximation. As Max Bisection is hard to approximate within α GW + ≈ 0.8786 under the Unique Games Conjecture (UGC), our algorithm is nearly optimal. We conjecture that Max Bisection is approximable within α GW − , i.e., that the bisection constraint (essentially) does not make Max Cut harder.We also obtain an optimal algorithm (assuming the UGC) for the analogous variant of Max 2-Sat. Our approximation ratio for this problem exactly matches the optimal approximation ratio for Max 2-Sat, i.e., α LLZ + ≈ 0.9401, showing that the bisection constraint does not make Max 2-Sat harder. This improves on a 0.93-approximation for this problem due to Raghavendra and Tan.

We show that, assuming the Unique Games Conjecture, it is NPhard to approximate MAX 2-SAT within α − LLZ + , where 0.9401 < α − LLZ < 0.9402 is the believed approximation ratio of the algorithm of Lewin, Livnat and Zwick [28].This result is surprising considering the fact that balanced instances of MAX 2-SAT, i.e., instances where each variable occurs positively and negatively equally often, can be approximated within 0.9439. In particular, instances in which roughly 68% of the literals are unnegated variables and 32% are negated appear less amenable to approximation than instances where the ratio is 50%-50%.