A characterization of strong approximation resistance

For a predicate f : {-1; 1}^κ →{0; 1} with ρ(f) = |f^-1(1)|/2^κ 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}^κ, 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.