Proceedings of the ninth annual ACM-SIAM symposium on Discrete algorithms
A PCP characterization of NP with optimal amortized query complexity
STOC '00 Proceedings of the thirty-second annual ACM symposium on Theory of computing
Some optimal inapproximability results
Journal of the ACM (JACM)
On the power of unique 2-prover 1-round games
STOC '02 Proceedings of the thiry-fourth annual ACM symposium on Theory of computing
A Tight Characterization of NP with 3 Query PCPs
FOCS '98 Proceedings of the 39th Annual Symposium on Foundations of Computer Science
Gowers uniformity, influence of variables, and PCPs
Proceedings of the thirty-eighth annual ACM symposium on Theory of computing
Near-optimal algorithms for unique games
Proceedings of the thirty-eighth annual ACM symposium on Theory of computing
Unique games on expanding constraint graphs are easy: extended abstract
STOC '08 Proceedings of the fortieth annual ACM symposium on Theory of computing
Every 2-csp Allows Nontrivial Approximation
Computational Complexity
On the Approximation Resistance of a Random Predicate
Computational Complexity
Spectral Algorithms for Unique Games
CCC '10 Proceedings of the 2010 IEEE 25th Annual Conference on Computational Complexity
Subexponential Algorithms for Unique Games and Related Problems
FOCS '10 Proceedings of the 2010 IEEE 51st Annual Symposium on Foundations of Computer Science
Beating the Random Ordering Is Hard: Every Ordering CSP Is Approximation Resistant
SIAM Journal on Computing
Approximating Linear Threshold Predicates
ACM Transactions on Computation Theory (TOCT)
A characterization of approximation resistance for even k-partite CSPs
Proceedings of the 4th conference on Innovations in Theoretical Computer Science
On the usefulness of predicates
ACM Transactions on Computation Theory (TOCT)
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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 $k\leq2$ 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 $c0$, a random predicate accepting $ct^2/\log t$ 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 $(32/33)\cdot2^t$ inputs is approximation resistant. The results extend from balanced distributions to arbitrary product distributions.