We show improved NP-hardness of approximating Ordering Constraint Satisfaction Problems (OCSPs). For the two most well-studied OCSPs, Maximum Acyclic Subgraph and Maximum Betweenness, we prove inapproximability of 14/15 + ε and 1/2 + ε.

An OCSP is said to be approximation resistant if it is hard to approximate better than taking a uniformly random ordering. We prove that the Maximum Non- Betweenness Problem is approximation resistant and that there are width-m approximation-resistant OCSPs accepting only a fraction 1 / (m/2)! of assignments. These results provide the first examples of approximation-resistant OCSPs subject only to PNP.

Our reductions from Label Cover differ from previous works in two ways. First, we establish a somewhat general bucketing lemma permitting us to reduce the analysis of ordering predicates to that of classical predicates. Second, instead of “folding”, which is not available for ordering predicates, we employ permuted instantiations of the predicates to limit the value of poorly correlated strategies.


Constraint Satisfaction Problem Acceptance Probability Inapproximability Result Approximation Resistance Label Cover 
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., Barak, B., Steurer, D.: Subexponential algorithms for Unique Games and related problems. In: FOCS, pp. 563–572. IEEE Computer Society (2010)Google Scholar
  2. 2.
    Arora, S., Lund, C., Motwani, R., Sudan, M., Szegedy, M.: Proof verification and the hardness of approximation problems. J. ACM 45(3), 501–555 (1998)MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Arora, S., Safra, S.: Probabilistic checking of proofs: A new characterization of NP. J. ACM 45(1), 70–122 (1998)MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Austrin, P., Mossel, E.: Approximation resistant predicates from pairwise independence. CCC 18(2), 249–271 (2009)MathSciNetMATHGoogle Scholar
  5. 5.
    Barak, B., Brandão, F.G.S.L., Harrow, A.W., Kelner, J.A., Steurer, D., Zhou, Y.: Hypercontractivity, sum-of-squares proofs, and their applications. In: Karloff, H.J., Pitassi, T. (eds.) STOC, pp. 307–326. ACM (2012)Google Scholar
  6. 6.
    Chan, S.O.: Approximation resistance from pairwise independent subgroups. In: STOC, pp. 325–337 (2013)Google Scholar
  7. 7.
    Charikar, M., Guruswami, V., Manokaran, R.: Every permutation CSP of arity 3 is approximation resistant. In: CCC, pp. 62–73 (2009)Google Scholar
  8. 8.
    Charikar, M., Makarychev, K., Makarychev, Y.: On the advantage over random for Maximum Acyclic Subgraph. In: FOCS, pp. 625–633. IEEE Computer Society (2007)Google Scholar
  9. 9.
    Chor, B., Sudan, M.: A geometric approach to betweenness. SIAM J. Disc. Math. 11(4), 511–523 (1998)MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Engebretsen, L., Holmerin, J.: More efficient queries in PCPs for NP and improved approximation hardness of maximum CSP. Rand. Struct. Algo. 33(4), 497–514 (2008)MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman & Co., New York (1979)MATHGoogle Scholar
  12. 12.
    Guruswami, V., Håstad, J., Manokaran, R., Raghavendra, P., Charikar, M.: Beating the random ordering is hard: Every ordering CSP is approximation resistant. SIAM J. Comput. 40(3), 878–914 (2011)MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Guruswami, V., Manokaran, R., Raghavendra, P.: Beating the random ordering is hard: Inapproximability of Maximum Acyclic Subgraph. In: FOCS, pp. 573–582 (2008)Google Scholar
  14. 14.
    Guruswami, V., Raghavendra, P., Saket, R., Wu, Y.: Bypassing UGC from some optimal geometric inapproximability results. In: Rabani, Y. (ed.) SODA, pp. 699–717. SIAM (2012)Google Scholar
  15. 15.
    Håstad, J.: Some optimal inapproximability results. J. ACM 48(4), 798–859 (2001)MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    Khot, S.: On the power of unique 2-prover 1-round games. In: STOC, pp. 767–775 (2002)Google Scholar
  17. 17.
    Mossel, E.: Gaussian bounds for noise correlation of functions. Geo. and Func. Anal. 19 (2010)Google Scholar
  18. 18.
    Newman, A.: The Maximum Acyclic Subgraph Problem and degree-3 graphs. In: Goemans, M.X., Jansen, K., Rolim, J.D.P., Trevisan, L. (eds.) APPROX-RANDOM 2001. LNCS, vol. 2129, pp. 147–158. Springer, Heidelberg (2001)Google Scholar
  19. 19.
    Raz, R.: A parallel repetition theorem. SIAM J. Comput. 27(3), 763–803 (1998)MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    Samorodnitsky, A., Trevisan, L.: A PCP characterization of NP with optimal amortized query complexity. In: Yao, F.F., Luks, E.M. (eds.) STOC, pp. 191–199. ACM (2000)Google Scholar
  21. 21.
    Trevisan, L., Sorkin, G.B., Sudan, M., Williamson, D.P.: Gadgets, approximation, and linear programming. SIAM J. Comput. 29(6), 2074–2097 (2000)MathSciNetMATHCrossRefGoogle Scholar
  22. 22.
    Wenner, C.: Circumventing d-to-1 for approximation resistance of satisfiable predicates strictly containing parity of width four (extended abstract). In: Gupta, A., Jansen, K., Rolim, J., Servedio, R. (eds.) APPROX/RANDOM 2012 LNCS, vol. 7408, pp. 325–337. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  23. 23.
    Wolff, P.: Hypercontractivity of simple random variables. Studia Mathematica, 219–326 (2007)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Per Austrin
    • 1
  • Rajsekar Manokaran
    • 1
  • Cenny Wenner
    • 1
  1. 1.School of Computer Science and CommunicationKTH – Royal Institute of TechnologyStockholmSweden

Personalised recommendations