Constraints, MMSNP and expander relational structures


We give a poly-time construction for a combinatorial classic known as Sparse Incomparability Lemma, studied by Erdős, Lovász, Nešetřil, Rödl and others: We show that every Constraint Satisfaction Problem is poly-time equivalent to its restriction to structures with large girth. This implies that the complexity classes CSP and Monotone Monadic Strict NP introduced by Feder and Vardi are computationally equivalent. The technical novelty of the paper is a concept of expander relations and a new type of product for relational structures: a generalization of the zig-zag product, the twisted product.

This is a preview of subscription content, access via your institution.


  1. [1]

    M. Ajtai, J. Komlós, E. Szemerédi: Sorting in c log n parallel steps, Combinatorica 3(1), (1983), 1–19.

    Article  MATH  MathSciNet  Google Scholar 

  2. [2]

    N. Alon, Oded Schwartz, Asaf Shapira: An elementary construction of constantdegree expanders, 17th ACM-SIAM Symposium on Discrete Algorithms (2007), 454–458.

    Google Scholar 

  3. [3]

    D. Duffus, V. Rödl, B. Sands, N. Sauer: Chromatic numbers and homomorphisms of large girth hypergraphs, preprint, (2006).

    Google Scholar 

  4. [4]

    P. Erdős: Graph theory and probability, Canad. J. Math. 11, (1959), 34–38.

    Article  MathSciNet  Google Scholar 

  5. [5]

    T. Feder, M. Y. Vardi: The computational structure of monotone monadic SNP and constraint satisfaction: A study through Datalog and group theory, SIAM J. Comput. 281 (1999), 57–104.

    MathSciNet  Google Scholar 

  6. [6]

    S. Hoory, N. Linial, A. Widgerson: Éxpander graphs and their applications,

  7. [7]

    A. Kostochka, J. Nešetřil, P. Smolíkova: Colorings and homomorphisms of bounded degree and degenerate graphs, Graph theory (Prague, 1998), Discrete Math. 233 (2001), no. 1–3, 257–276.

    Article  MATH  MathSciNet  Google Scholar 

  8. [8]

    G. Kun: On the complexity of Constraint Satisfaction Problem, PhD thesis (in Hungarian), 2006.

    Google Scholar 

  9. [9]

    G. Kun, J. Nešetřil: Forbidden lifts (NP and CSP for combinatorists), KAMDIMATIA Series 2006-775 (to appear in European J. Comb.).

  10. [10]

    L. Lovász: On chromatic number of finite set-systems, Acta Math. Acad. Sci. Hungar. 19 1968, 59–67.

    Article  MATH  MathSciNet  Google Scholar 

  11. [11]

    A. Lubotzky, R. Phillips, P. Sarnak: Ramanujan graphs, Combinatorica 8(3) 261–277, 1988.

    Article  MATH  MathSciNet  Google Scholar 

  12. [12]

    A. Lubotzky, B. Samuels, V. Vishne: Ramanujan complexes of type A d, Israel J. of Math. 2005, accepted.

    Google Scholar 

  13. [13]

    A. Lubotzky, B. Samuels, V. Vishne: Explicit constructions of Ramanujan complexes of type A d, Europ. J. of Combinatorics 2005, submitted.

    Google Scholar 

  14. [14]

    G. A. Margulis: Explicit group-theoretical constructions of combinatorial schemes and their application to the design of expanders and concentrators, J. Probl. Inf. Transm. 24, (1988), 39–46.

    MATH  MathSciNet  Google Scholar 

  15. [15]

    J. Matoušek, J. Nešetřil: Constructions of sparse graphs with given homomorphisms (to appear).

  16. [16]

    J. Nešetřil, V. Rödl: A short proof of the existence of highly chromatic hypergraphs without short cycle, J. Comb. Th. B 27 (1979), 225–227.

    Article  MATH  Google Scholar 

  17. [17]

    J. Nešetřil, M. H. Siggers: A new combinatorial approach to the Constraint Satisfaction Problem dichotomy conjecture, 32nd Symposium on the Mathematical Foundation of Computer Science, 2007, submitted.

    Google Scholar 

  18. [18]

    O. Reingold, S. Vadhan, A. Widgerson: Entropy, waves the zig-zag product, and new constant degree expanders, Annals of mathematics 155(1), (2002), 157–187.

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information



Corresponding author

Correspondence to Gábor Kun.

Additional information

This research was supported by OTKA Grants no. T043671 and NK 67867, Subhash Khot’s NSF Waterman Award CCF-1061938 and the MTA Rényi “Lendület” Groups and Graphs Research Group.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Kun, G. Constraints, MMSNP and expander relational structures. Combinatorica 33, 335–347 (2013).

Download citation

Mathematics Subject Classification (2000)

  • Fill in
  • please