Showing papers by "Johan Håstad published in 2009"

Randomly supported independence and resistance

Abstract: We prove that for any positive integer k, there is a constant ck such that a randomly selected set of ck nk log n Boolean vectors with high probability supports a balanced k-wise independent distribution. In the case of k ≤ 2 a more elaborate argument gives the stronger bound ck nk. Using a recent result by Austrin and Mossel this shows that a predicate on t bits, chosen at random among predicates accepting c2 t2 input vectors, is, assuming the Unique Games Conjecture, likely to be approximation resistant. These results are close to tight: we show that there are other constants, ck', such that a randomly selected set of cardinality ck' nk points is unlikely to support a balanced k-wise independent distribution and, for some c>0, a random predicate accepting ct2/log t input vectors is non-trivially approximable with high probability. In a different application of the result of Austrin and Mossel we prove that, again assuming the Unique Games Conjecture, any predicate on t bits accepting at least (32/33) • 2t inputs is approximation resistant. The results extend from the Boolean domain to larger finite domains.

On the Approximation Resistance of a Random Predicate

Abstract: A predicate is called approximation resistant if it is NP-hard to approximate the corresponding constraint satisfaction problem significantly better than what is achieved by the naive algorithm that picks an assignment uniformly at random. In this paper we study predicates of Boolean inputs where the width of the predicate is allowed to grow. Samorodnitsky and Trevisan proved that, assuming the Unique Games Conjecture, there is a family of very sparse predicates that are approximation resistant. We prove that, under the same conjecture, any predicate implied by their predicate remains approximation resistant and that, with high probability, this condition applies to a randomly chosen predicate.