A theorem of Håstad shows that for every constraint satisfaction problem (CSP) over a fixed size domain, instances where each variable appears in at most O(1) constraints admit a non-trivial approximation algorithm, in the sense that one can beat (by an additive constant) the approximation ratio achieved by the naive algorithm that simply picks a random assignment. We consider the analogous question for ordering CSPs, where the goal is to find a linear ordering of the variables to maximize the number of satisfied constraints, each of which stipulates some restriction on the local order of the involved variables. It was shown recently that without the bounded occurrence restriction, for every ordering CSP it is Unique Games-hard to beat the naive random ordering algorithm.

In this work, we prove that the CSP with monotone ordering constraints \(x_{i_1} < x_{i_2} < \cdots < x_{i_k}\) of arbitrary arity k can be approximated beyond the random ordering threshold 1/k! on bounded occurrence instances. We prove a similar result for all ordering CSPs, with arbitrary payoff functions, whose constraints have arity at most 3. Our method is based on working with a carefully defined Boolean CSP that serves as a proxy for the ordering CSP. One of the main technical ingredients is to establish that certain Fourier coefficients of this proxy constraint have substantial mass. These are then used to guarantee a good ordering via an algorithm that finds a good Boolean assignment to the variables of a low-degree bounded occurrence multilinear polynomial. Our algorithm for the latter task is similar to Håstad’s earlier method but is based on a greedy choice that achieves a better performance guarantee.


Constraint Satisfaction Prob Naive Algorithm Boolean Cube Constraint Satisfaction Prob Instance Nontrivial Approximation 
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. [AM09]
    Austrin, P., Mossel, E.: Approximation resistant predicates from pairwise independence. Computational Complexity 18(2), 249–271 (2009); Preliminary version in CCC 2008MathSciNetzbMATHCrossRefGoogle Scholar
  2. [BK10]
    Bodirsky, M., Kára, J.: The complexity of temporal constraint satisfaction problems. J. ACM 57, 9:1–9:41 (2010)Google Scholar
  3. [BS97]
    Berger, B., Shor, P.W.: Tight bounds for the Maximum Acyclic Subgraph problem. J. Algorithms 25(1), 1–18 (1997)MathSciNetzbMATHCrossRefGoogle Scholar
  4. [CGM09]
    Charikar, M., Guruswami, V., Manokaran, R.: Every permutation CSP of arity 3 is approximation resistant. In: Proceedings of the 24th IEEE Conference on Computational Complexity, pp. 62–73 (July 2009)Google Scholar
  5. [CS98]
    Chor, B., Sudan, M.: A geometric approach to betweenness. SIAM J. Discrete Math. 11(4), 511–523 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  6. [EG04]
    Engebretsen, L., Guruswami, V.: Is constraint satisfaction over two variables always easy? Random Structures and Algorithms 25(2), 150–178 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  7. GHM+11.
    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)MathSciNetzbMATHCrossRefGoogle Scholar
  8. [GM06]
    Guttmann, W., Maucher, M.: Variations on an Ordering Theme with Constraints. In: Navarro, G., Bertossi, L.E., Kohayakawa, Y. (eds.) IFIP TCS. IFIP, vol. 209, pp. 77–90. Springer, Boston (2006)CrossRefGoogle Scholar
  9. [GMR08]
    Guruswami, V., Manokaran, R., Raghavendra, P.: Beating the random ordering hard: Inapproximability of maximum acyclic subgraph. In: Proceedings of the 49th IEEE Symposium on Foundations of Computer Science, pp. 573–582 (2008)Google Scholar
  10. [GW95]
    Goemans, M.X., Williamson, D.P.: Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming. Journal of the ACM 42(6), 1115–1145 (1995)MathSciNetzbMATHCrossRefGoogle Scholar
  11. [Hås00]
    Håstad, J.: On bounded occurrence constraint satisfaction. Inf. Process. Lett. 74(1-2), 1–6 (2000)zbMATHCrossRefGoogle Scholar
  12. [Hås01]
    Håstad, J.: Some optimal inapproximability results. Journal of the ACM 48(4), 798–859 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  13. [Hås08]
    Håstad, J.: Every 2-CSP allows nontrivial approximation. Computational Complexity 17(4), 549–566 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  14. [Mak09]
    Makarychev, Y.: Simple linear time approximation algorithm for betweenness. Microsoft Research Technical Report MSR-TR-2009-74 (2009)Google Scholar
  15. [Rag08]
    Raghavendra, P.: Optimal algorithms and inapproximability results for every CSP? In: Proceedings of the 40th ACM Symposium on Theory of Computing, pp. 245–254 (2008)Google Scholar
  16. [Tre01]
    Trevisan, L.: Non-approximability results for optimization problems on bounded degree instances. In: Proceedings of the 33rd Annual ACM Symposium on Theory of Computing, pp. 453–461 (2001)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Venkatesan Guruswami
    • 1
  • Yuan Zhou
    • 1
  1. 1.Computer Science DepartmentCarnegie Mellon UniversityUSA

Personalised recommendations