UG-hardness to NP-hardness by Losing Half.

The 2-to-2 Games Theorem of [16, 10, 11, 17] implies that it is NP-hard to distinguish between Unique Games instances with assignment satisfying at least (1/2 - epsilon) fraction of the constraints vs. no assignment satisfying more than epsilon fraction of the constraints, for every constant epsilon > 0. We show that the reduction can be transformed in a non-trivial way to give a stronger guarantee in the completeness case: For at least (1/2 - epsilon) fraction of the vertices on one side, all the constraints associated with them in the Unique Games instance can be satisfied. We use this guarantee to convert the known UG-hardness results to NP-hardness. We show: 1. Tight inapproximability of approximating independent sets in degree d graphs within a factor of Omega(d/log(2) d), where d is a constant. 2. NP-hardness of approximate the Maximum Acyclic Subgraph problem within a factor of 2/3 + epsilon, improving the previous ratio of 14/15 + epsilon by Austrin et al. [4]. 3. For any predicate P-1(1) subset of [q](k) supporting a balanced pairwise independent distribution, given a P-CSP instance with value at least 1/2 - epsilon, it is NP-hard to satisfy more than vertical bar P-1(1)vertical bar/q(k) + epsilon fraction of constraints.