We prove that for any positive integer k, there is a constant c k such that a randomly selected set of c k n k 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 c k n k . Using a recent result by Austrin and Mossel this shows that a predicate on t bits, chosen at random among predicates accepting c2t 2 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, c k , such that a randomly selected set of cardinality c k n k points is unlikely to support a balanced kwise independent distribution and, for some c > 0, a random predicate accepting ct 2 / 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) · 2 t inputs is approximation resistant. The results extend from the Boolean domain to larger finite domains.
In a subsequent work by REF , the results of this paper were used to derive strong lower bounds on the probability that a random predicate is resistant.
Randomly supported independence and resistance
STOC '09
Mathematics Computer Science