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 (¹₂ − ε) fraction of the constraints vs. no assignment satisfying more than ε fraction of the constraints, for every constant ε > 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 (¹₂ − ε) 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 Ω _log^d_{2 d} , where d is a constant. 2. NP-hardness of approximate the Maximum Acyclic Subgraph problem within a factor of ²₃ + ε, improving the previous ratio of ¹⁴₁₅ + ε by Austrin et al. [4]. 3. For any predicate P⁻¹(1) ⊆ [q]^k supporting a balanced pairwise independent distribution, given a P-CSP instance with value at least ¹₂ − ε, it is NP-hard to satisfy more than ^|P−^1(1)| + ε fraction of constraints. q^k.