Max CSP( P ) is the problem of maximizing the weight of satisfied constraints, where each constraint acts over a k-tuple of literals and is evaluated using the predicate P. The approximation ratio of a random assignment is equal to the fraction of satisfying inputs to P. If it is NP-hard to achieve a better approximation ratio for Max CSP( P ), then we say that P is approximation resistant. Our goal is to characterize which predicates that have this property.

A general approximation algorithm for Max CSP( P ) is introduced. For a multitude of different P, it is shown that the algorithm beats the random assignment algorithm, thus implying that P is not approximation resistant. In particular, over 2/3 of the predicates on four binary inputs are proved not to be approximation resistant, as well as all predicates on 2s binary inputs, that have at most 2s+1 accepting inputs.

We also prove a large number of predicates to be approximation resistant. In particular, all predicates of arity 2s+s 2 with less than \(2^{s^2}\) non-accepting inputs are proved to be approximation resistant, as well as almost 1/5 of the predicates on four binary inputs.


Random Assignment Constraint Satisfaction Problem Satisfying Assignment Annual IEEE Symposium Hard Instance 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Arora, S., Lund, C., Motwani, R., Sudan, M., Szegedy, M.: Proof verification and hardness of approximation problems. Journal of the ACM 45(3), 501–555 (1998); Preliminary version appeared in FOCS 1992zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Bellare, M., Goldreich, O., Sudan, M.: Free bits, PCPs, and nonapproximability - towards tight results. SIAM Journal on Computing 27(3), 804–915 (1998)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Charikar, M., Wirth, A.: Maximizing quadratic programs: Extending Grothendieck’s inequality. In: Proceedings of the 45th Annual IEEE Symposium on Foundations of Computer Science, pp. 54–60 (2004)Google Scholar
  4. 4.
    Creignou, N.: A dichotomy theorem for maximum generalized satisability problems. Journal of Computer and System Sciences 51(3), 511–522 (1995)CrossRefMathSciNetGoogle Scholar
  5. 5.
    Engebretsen, L., Holmerin, J.: More efficient queries in PCPs for NP and improved approximation hardness of maximum CSP. In: Diekert, V., Durand, B. (eds.) STACS 2005. LNCS, vol. 3404, pp. 194–205. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  6. 6.
    Feige, U.: A threshold of ln n for approximating set cover. Journal of the ACM 45(4), 634–652 (1998)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Feige, U.: Relations between average case complexity and approximation complexity. In: Proceedings of the 34th Annual ACM Symposium on Theory of Computing, pp. 534–543 (2002)Google Scholar
  8. 8.
    Feige, U., Goldwasser, S., Lovász, L., Safra, S., Szegedy, M.: Interactive proofs and the hardness of approximating cliques. Journal of the ACM 43(2), 268–292 (1996); Preliminary version appeared in FOCS 1991zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Feige, U., Langberg, M.: The RPR2 rounding technique for semidefinite programs. In: Orejas, F., Spirakis, P.G., van Leeuwen, J. (eds.) ICALP 2001. LNCS, vol. 2076, pp. 213–224. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  10. 10.
    Goemans, M.X., Williamson, D.P.: Improved Approximation Algorithms for Maximum Cut and Satisfiability Problems Using Semidefinite Programming. Journal of the ACM 42, 1115–1145 (1995)zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Guruswami, V., Lewin, D., Sudan, M., Trevisan, L.: A tight characterization of NP with 3 query PCPs. In: Proceedings of the 39th Annual IEEE Symposium on Foundations of Computer Science, pp. 8–17 (1998)Google Scholar
  12. 12.
    Hast, G.: Beating a Random Assignment. PhD thesis, Royal Institute of Technology (2005)Google Scholar
  13. 13.
    Hăstad, J.: Some optimal inapproximability results. Journal of the ACM 48(4), 798–859 (1997); Preliminary version appeared in STOC 1997CrossRefGoogle Scholar
  14. 14.
    Hăstad, J.: Every 2-CSP allows nontrivial approximation. In: Proceedings of the 37th Annual ACM Symposium on Theory of Computing, pp. 740–746 (2005)Google Scholar
  15. 15.
    Hăstad, J., Wigderson, A.: Simple analysis of graph tests for linearity and PCP. Random Structures and Algorithms 22(2), 139–160 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Karloff, H., Zwick, U.: A 7/8-approximation algorithm for MAX 3SAT. In: Proceedings of the 38th Annual IEEE Symposium on Foundations of Computer Science, pp. 406–415 (1997)Google Scholar
  17. 17.
    Khanna, S., Sudan, M., Trevisan, L., Williamson, D.P.: The approximability of constraint satisfaction problems. SIAM Journal on Computing 30(6), 1863–1920 (2000)CrossRefMathSciNetGoogle Scholar
  18. 18.
    Lewin, M., Livnat, D., Zwick, U.: Improved rounding techniques for the MAX 2-SAT and MAX DI-CUT problems. In: Cook, W.J., Schulz, A.S. (eds.) IPCO 2002. LNCS, vol. 2337, pp. 67–82. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  19. 19.
    Lund, C., Yannakakis, M.: On the hardness of approximating minimization problems. Journal of the ACM 41(5), 960–981 (1994); Preliminary version appeared in STOC 1993zbMATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Papadimitriou, C.H., Yannakakis, M.: Optimization, approximation, and complexity classes. Journal of Computer and System Sciences 43(3), 425–440 (1991)zbMATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Samorodnitsky, A., Trevisan, L.: A PCP characterization of NP with optimal amortized query complexity. In: Proceedings of the 32nd Annual ACM Symposium on Theory of Computing, pp. 191–199 (2000)Google Scholar
  22. 22.
    Zwick, U.: Approximation algorithms for constraint satisfaction problems involving at most three variables per constraint. In: Proceedings of the 9th Annual ACMSIAM Symposium on Discrete Algorithms, pp. 201–210 (1998)Google Scholar
  23. 23.
    Zwick, U.: Outward rotations: a tool for rounding solutions of semidefinite programming relaxations, with applications to MAX CUT and other problems. In: Proceedings of the 31st Annual ACM Symposium on Theory of Computing, pp. 679–687 (1999)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Gustav Hast
    • 1
  1. 1.Department of Numerical Analysis and Computer ScienceRoyal Institute of TechnologyStockholmSweden

Personalised recommendations