We prove that, for any ε > 0, it is NP-hard to, given a satisfiable instance of Max-NTW (Not-2), find an assignment that satisfies a fraction \(\frac 58 +\epsilon\) of the constraints. This, up to the existence of ε, matches the approximation ratio obtained by the trivial algorithm that just picks an assignment at random and thus the result is tight. Said equivalently the result proves that Max-NTW is approximation resistant on satisfiable instances and this makes our understanding of arity three Max-CSPs with regards to approximation resistance complete.


Constraint Satisfaction Problem Correlate Space Extended Game Approximation Resistance Label Cover 
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  1. 1.
    Arora, S., Lund, C., Motwani, R., Sudan, M., Szegedy, M.: Proof verification and intractability of approximation problems. Journal of the ACM 45, 501–555 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Austrin, P., Håstad, J.: Randomly supported independence and resistance. SIAM Journal on Computing 40, 1–27 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Austrin, P., Håstad, J.: On the usefulness of predicates. To appear at the Conference for Computational Complexity, 2012 (2012)Google Scholar
  4. 4.
    Goemans, M., Williamson, D.: Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming. Journal of the ACM 42, 1115–1145 (1995)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Håstad, J.: Clique is hard to approximate within n 1 − ε. Acta Mathematica 182, 105–142 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Håstad, J.: Some optimal inapproximability results. JACM 48, 798–859 (2001)zbMATHCrossRefGoogle Scholar
  7. 7.
    Håstad, J.: Satisfying Degree-d Equations over GF[2]n. In: Goldberg, L.A., et al. (eds.) RANDOM 2011 and APPROX 2011. LNCS, vol. 6845, pp. 242–253. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  8. 8.
    Huang, S.: Approximation resistance on satisfiable instances for predicates strictly dominating parity. ECCC Report 12-040 (2012)Google Scholar
  9. 9.
    Khot, S.: On the power of unique 2-prover 1-round games. In: Proceedings of 34th ACM Symposium on Theory of Computating, pp. 767–775 (2002)Google Scholar
  10. 10.
    Khot, S., Saket, R.: A 3-query non-adaptive pcp with perfect completeness. In: Proc. of 21st Annual Conference on Computational, pp. 159–169. IEEE Computer Society (2006)Google Scholar
  11. 11.
    Khot, S.: Hardness results for coloring 3-colorable 3-uniform hypergraphs. In: Proceedings of 43rd Annual IEEE Symposium of Foundations of Computer Science, pp. 23–32 (2002)Google Scholar
  12. 12.
    Mossel, E.: Gaussian bounds for noise correlation of functions. GAFA 19, 1713–1756 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    O’Donnell, R., Wu, Y.: Conditional hardness for satisfiable 3-CSPs. In: Proceedings of 41st ACM Symposium on Theory of Computating, pp. 493–502 (2009)Google Scholar
  14. 14.
    Raz, R.: A parallel repetition theorem. SIAM J. on Computing 27, 763–803 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Wenner, C.: Circumventing D-to-1 for Approximation Resistance of Satisfiable Predicates Strictly Containing Parity of Width Four. In: Gupta, A., et al. (eds.) APPROX/RANDOM 2012. LNCS, vol. 7408, pp. 325–337. Springer, Heidelberg (2012)Google Scholar
  16. 16.
    Zwick, U.: Approximation algorithms for constraint satisfaction problems involving at most three variables per constraint. In: Proceedings 9th Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 201–210. ACM (1998)Google Scholar

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© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Johan Håstad
    • 1
  1. 1.KTH Royal Institute of TechnologySweden

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