computational complexity

, Volume 18, Issue 2, pp 249–271 | Cite as

Approximation Resistant Predicates from Pairwise Independence



We study the approximability of predicates on k variables from a domain [q], and give a new sufficient condition for such predicates to be approximation resistant under the Unique Games Conjecture. Specifically, we show that a predicate P is approximation resistant if there exists a balanced pairwise independent distribution over [q] k whose support is contained in the set of satisfying assignments to P. Using constructions of pairwise independent distributions this result implies that
  • For general k ≥ 3 and q ≥ 2, the Max k-CSP q problem is UG-hard to approximate within \({\mathcal{O}}(kq^2)/q^k + \epsilon\).

  • For the special case of q = 2, i.e., boolean variables, we can sharpen this bound to \((k +{\mathcal{O}}(k^{0.525}))/2^k +\epsilon\), improving upon the best previous bound of 2k/2 k  + ∈ (Samorodnitsky and Trevisan, STOC’06) by essentially a factor 2.

  • Finally, again for q = 2, assuming that the famous Hadamard Conjecture is true, this can be improved even further, and the \({\mathcal{O}}(k^{0.525})\) term can be replaced by the constant 4.


Approximation resistance constraint satisfaction unique games conjecture 

Subject classification.

MSC Primary 68Q17 Secondary 41A52 


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Copyright information

© Birkhäuser Verlag, Basel 2009

Authors and Affiliations

  1. 1.KTH - Royal Institute of TechnologyStockholmSweden
  2. 2.U.C. BerkeleyBerkeleyUSA
  3. 3.The Weizmann Institute of ScienceRehovotIsrael

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