A characterization of strong approximation resistance

For a predicate f: {-1, 1}k ↦ {0, 1} with ρ(f) = |f-1(1)|/2k, we call the predicate strongly approximation resistant if given a near-satisfiable instance of CSP(f), it is computationally hard to find an assignment such that the fraction of constraints satisfied is outside the range [ρ(f) - Ω(1), ρ(f) + Ω(1)]. We present a characterization of strongly approximation resistant predicates under the Unique Games Conjecture. We also present characterizations in the mixed linear and semi-definite programming hierarchy and the Sherali-Adams linear programming hierarchy. In the former case, the characterization coincides with the one based on UGC. Each of the two characterizations is in terms of existence of a probability measure on a natural convex polytope associated with the predicate. The predicate is called approximation resistant if given a near-satisfiable instance of CSP(f), it is computationally hard to find an assignment such that the fraction of constraints satisfied is at least ρ(f) + Ω(1). When the predicate is odd, i.e. f(-z) = 1 - f(z), ∀z ∈ {-1, 1}k, it is easily observed that the notion of approximation resistance coincides with that of strong approximation resistance. Hence for odd predicates our characterization of strong approximation resistance is also a characterization of approximation resistance.