Parity is Positively Useless

We give the first examples of non-trivially positively-useless predicates subject only to P 6= NP. In particular, for every constraint function Q : {−1, 1}4 → R, we construct Contraint-SatisfactionProblem (CSP) instances without negations which have value at least 1 − ε when evaluted for the arity-four odd-parity predicate, yet it is NP-hard to find a solution with value significantly better than a random biased assignment when evaluated for Q. More generally, we show that all parities except one are positively useless. Although we are not able to exhibit a single protocol producing hard instances when evaluated for every Q, we show that two protocols do the trick. The first protocol is the classical one used by Håstad with a twist. We extend the protocol to multilayered Label Cover and employ a particular distribution over layers in order to limit moments of table biases. The second protocol is a modification of Chan’s multi-question protocol where queried tuples of Label Cover vertices are randomized in such a way that the tables can be seen as being independently sampled from a common distribution and in effect having identical expected biases. We believe that our techniques may prove useful in further analyzing the approximability of CSPs without negations. 1998 ACM Subject Classification F.2.2 Nonnumerical Algorithms and Problems