Randomly supported independence and resistance

We prove that for any positive integer k, there is a constant ck such that a randomly selected set of cknk 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 cknk . Using a recent result by Austrin and Mossel this shows that a predicate on t bits, chosen at random among predicates accepting c2t2 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 cknk points is unlikely to support a balanced kwise 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 acceptin...

• Type: Conference
• Year: 2009
• Where: STOC
• Authors: Per Austrin, Johan Håstad
• Keywords: Algorithms, Boolean Vectors, Input Vectors, Random Predicate, STOC 2009