  A predicate f:{-1,1}^k -> {0,1} with \rho(f) = \frac{|f^{-1}(1)|}{2^k} is
called {\it approximation resistant} if given a near-satisfiable instance of
CSP(f), it is computationally hard to find an assignment that satisfies at
least \rho(f)+\Omega(1) fraction of the constraints.
  We present a complete characterization of approximation resistant predicates
under the Unique Games Conjecture. We also present characterizations in the
{\it 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 with
certain symmetry properties on a natural convex polytope associated with the
predicate.
