Randomly supported independence and resistance
2011 (English)In: SIAM journal on computing (Print), ISSN 0097-5397, E-ISSN 1095-7111, Vol. 40, no 1, 1-27 p.Article in journal (Refereed) Published
We prove that for any positive integers q and k there is a constant c(q,k) such that a uniformly random set of c(q,k)n(k) log n vectors in [q](n) 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(q,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 c(q,2)t(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'(q,k), such that a randomly selected set of cardinality c'(q,k)n(k) points is unlikely to support a balanced k-wise independent distribution and, for some c > 0, a random predicate accepting ct(2)/logt input vectors is nontrivially 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 Boolean inputs accepting at least (32/33).2(t) inputs is approximation resistant. The results extend from balanced distributions to arbitrary product distributions.
Place, publisher, year, edition, pages
2011. Vol. 40, no 1, 1-27 p.
k-wise independence, constraint satisfaction, approximation resistance
IdentifiersURN: urn:nbn:se:kth:diva-31300DOI: 10.1137/100783534ISI: 000287697400001ScopusID: 2-s2.0-79952948997OAI: oai:DiVA.org:kth-31300DiVA: diva2:405153
FunderSwedish Research Council, 50394001
QC 201506292011-03-212011-03-142015-06-29Bibliographically approved