Proving Weak Approximability Without Algorithms

A predicate f : {−1, 1} 7→ {0, 1} with ρ(f) = Ex∈{−1,1}k [f(x)] is said to be strongly approximation resistant if, for every ε > 0, given a near-satisfiable instance of MAX k-CSP(f), it is hard to find an assignment such that the fraction of constraints satisfied is outside the range [ρ(f)−ε, ρ(f) +ε]. A predicate which is not strongly approximation resistant is known as weakly approximable. We give a new method for proving the weak approximability of predicates, using a simple SDP relaxation, without designing and analyzing new rounding algorithms for each predicate. Instead, we use the recent characterization of strong approximation resistance by Khot et. al [13], and show how to prove that for a given predicate f , certain necessary conditions for strong resistance derived from their characterization, are violated. By their result, this implies the existence of a good rounding algorithm, proving weak approximability. We show how this method can be used to obtain simple proofs of (weak approximability analogues of) various known results on approximability, as well as new results on weak approximability of symmetric predicates. 1998 ACM Subject Classification F.2 Analysis of Algorithms and Problem Complexity