Advertisement

Approximability of the Maximum Solution Problem for Certain Families of Algebras

  • Peter Jonsson
  • Johan Thapper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5675)

Abstract

We study the approximability of the maximum solution problem. This problem is an optimisation variant of the constraint satisfaction problem and it captures a wide range of interesting problems in, for example, integer programming, equation solving, and graph theory. The approximability of this problem has previously been studied in the two-element case [Khanna et al, ‘The approximability of constraint satisfaction’, SIAM Journal on Computing 23(6), 2000] and in some algebraically motivated cases [Jonsson et al, ‘Max Ones generalized to larger domains’, SIAM Journal on Computing 38(1), 2008]. We continue this line of research by considering the approximability of Max Sol for different types of constraints. Our investigation combined with the older results strengthens the hypothesis that Max Sol exhibits a pentachotomy with respect to approximability.

Keywords

Optimisation approximability constraint satisfaction algebra computational complexity 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Ausiello, G., Crescenzi, P., Gambosi, G., Kann, V., Marchetti-Spaccamela, A., Protasi, M.: Complexity and Approximation. Springer, Heidelberg (1999)CrossRefzbMATHGoogle Scholar
  2. 2.
    Bulatov, A.: Combinatorial problems raised from 2-semilattices. Journal of Algebra 298, 321–339 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bulatov, A., Jeavons, P., Krokhin, A.: Classifying the computational complexity of constraints using finte algebras. SIAM Journal on Computing 34(3), 720–742 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Goldman, J., Rota, G.-C.: On the foundations of combinatorial theory IV. Finite vector spaces and Eulerian generating functions. Studies in Appl. Math. 49, 239–258 (1970)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Jeavons, P.: On the algebraic structure of combinatorial problems. Theoretical Computer Science 200(1-2), 185–204 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Jeavons, P., Cohen, D., Gyssens, M.: Closure properties of constraints. Journal of the ACM 44(4), 527–548 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Jonsson, P., Kuivinen, F., Nordh, G.: Max Ones generalized to larger domains. SIAM Journal on Computing 38(1), 329–365 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Jonsson, P., Nordh, G.: Generalised integer programming based on logically defined relations. In: Královič, R., Urzyczyn, P. (eds.) MFCS 2006. LNCS, vol. 4162, pp. 549–560. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  9. 9.
    Jonsson, P., Nordh, G., Thapper, J.: The maximum solution problem on graphs. In: Kučera, L., Kučera, A. (eds.) MFCS 2007. LNCS, vol. 4708, pp. 228–239. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  10. 10.
    Khanna, S., Sudan, M., Trevisan, L., Williamson, D.P.: The approximability of constraint satisfaction problems. SIAM Journal on Computing 30(6), 1863–1920 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Kuivinen, F.: Tight approximability results for the maximum solution equation problem over Z p. In: Jedrzejowicz, J., Szepietowski, A. (eds.) MFCS 2005. LNCS, vol. 3618, pp. 628–639. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  12. 12.
    Pöschel, R., Kalužnin, L.: Funktionen- und Relationenalgebren. DVW, Berlin (1979)CrossRefGoogle Scholar
  13. 13.
    Post, E.: The two-valued iterative systems of mathematical logic. Annals of Mathematical Studies 5, 1–122 (1941)MathSciNetGoogle Scholar
  14. 14.
    Szendrei, Á.: Clones in Universal Algebra. Seminaires de Mathématiques Supérieures, vol. 99. University of Montreal (1986)Google Scholar
  15. 15.
    Szendrei, Á.: Symmetric algebras. In: Contributions to General Algebra 6, pp. 259–280. Verlag Hölder-Pichler-Tempsky, Wien and Verlag Teubner, Stuttgart (1989)Google Scholar
  16. 16.
    Szendrei, Á.: Simple surjective algebras having no proper subalgebras. J. Austral. Math. Soc. ser. A 48, 434–454 (1990)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Peter Jonsson
    • 1
  • Johan Thapper
    • 1
  1. 1.Department of Computer and Information ScienceLinköpings universitetLinköpingSweden

Personalised recommendations