Skip to main content
SpringerLink
Log in

Tractable Languages

  • Chapter
The Complexity of Valued Constraint Satisfaction Problems

Part of the book series: Cognitive Technologies ((COGTECH))

  • 598 Accesses

Abstract

Chapter 6 surveys a list of known tractable valued constraint languages. Quite strikingly, all of them can be characterised by simple fractional polymorphisms.

I learned very early the difference between knowing the name of something and knowing something.

Richard Feynman

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    Think of 3V as {0,1,2}-vectors of length |V|.

  2. 2.

    It is easy to show that an equivalent definition of k-submodularity is a generalisation of submodularity and bisubmodularity in terms of two k-tuples of pairwise disjoint subsets of a given finite set.

  3. 3.

    A chain is a binary tree in which all nodes except leaves have exactly one child.

  4. 4.

    A fork is a binary tree in which all nodes except leaves and one special node have exactly one child. The special node has exactly two children.

  5. 5.

    A ternary operation f:D 3→D is called Mal’tsev if f(x,y,y)=f(y,y,x)=x for all x,y∈D.

References

  1. Barto, L., Kozik, M.: Constraint satisfaction problems of bounded width. In: Proceedings of the 50th Annual IEEE Symposium on Foundations of Computer Science (FOCS’09), pp. 461–471. IEEE Computer Society, Los Alamitos (2009)

    Google Scholar 

  2. Bilbao, J.M., Fernández, J.R., Jiménez, N., López, J.J.: Survey of bicooperative games. In: Chinchuluun, A., Pardalos, P.M., Migdalas, A., Pitsoulis, L. (eds.) Pareto Optimality, Game Theory and Equilibria. Springer, Berlin (2008)

    Google Scholar 

  3. Bouchet, A.: Greedy algorithm and symmetric matroids. Math. Program. 38(1), 147–159 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bulatov, A.: A dichotomy theorem for constraint satisfaction problems on a 3-element set. J. ACM 53(1), 66–120 (2006)

    MathSciNet  Google Scholar 

  5. Bulatov, A., Dalmau, V.: A simple algorithm for Mal’tsev constraints. SIAM J. Sci. Comput. 36(1), 16–27 (2006)

    MathSciNet  MATH  Google Scholar 

  6. Chandrasekaran, R., Kabadi, S.N.: Pseudomatroids. Discrete Math. 71(3), 205–217 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  7. Cohen, D., Jeavons, P.: The complexity of constraint languages. In: Rossi, F., van Beek, P., Walsh, T. (eds.) The Handbook of Constraint Programming. Elsevier, Amsterdam (2006)

    Google Scholar 

  8. Cohen, D.A., Cooper, M.C., Jeavons, P.G.: Generalising submodularity and Horn clauses: tractable optimization problems defined by tournament pair multimorphisms. Theor. Comput. Sci. 401(1–3), 36–51 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  9. Cohen, D.A., Cooper, M.C., Jeavons, P.G., Krokhin, A.A.: The complexity of soft constraint satisfaction. Artif. Intell. 170(11), 983–1016 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  10. Cooper, M.C.: Minimization of locally defined submodular functions by optimal soft arc consistency. Constraints 13(4), 437–458 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  11. Cooper, M.C., de Givry, S., Sánchez, M., Schiex, T., Zytnicki, M., Werner, T.: Soft arc consistency revisited. Artif. Intell. 174(7–8), 449–478 (2010)

    Article  MATH  Google Scholar 

  12. Creed, P., Živný, S.: On minimal weighted clones. In: Proceedings of the 17th International Conference on Principles and Practice of Constraint Programming (CP’11). Lecture Notes in Computer Science, vol. 6876, pp. 210–224. Springer, Berlin (2011)

    Google Scholar 

  13. Dalmau, V.: Generalized majority-minority operations are tractable. Log. Methods Comput. Sci. 2(4) (2006)

    Google Scholar 

  14. Fujishige, S., Iwata, S.: Bisubmodular function minimization. SIAM J. Discrete Math. 19(4), 1065–1073 (2005)

    Article  MathSciNet  Google Scholar 

  15. Huber, A., Kolmogorov, V.: Towards minimizing k-submodular functions. In: Proceedings of the 2nd International Symposium on Combinatorial Optimization (ISCO’12) (2012)

    Google Scholar 

  16. Huber, A., Krokhin, A., Powell, R.: Skew bisubmodularity and valued CSPs. In: Proceedings of the 24th ACM-SIAM Symposium on Discrete Algorithms (SODA’13). SIAM, Philadelphia (2012)

    Google Scholar 

  17. Idziak, P.M., Markovic, P., McKenzie, R., Valeriote, M., Willard, R.: Tractability and learnability arising from algebras with few subpowers. SIAM J. Comput. 39(7), 3023–3037 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  18. Iwata, S., Fleischer, L., Fujishige, S.: A combinatorial strongly polynomial algorithm for minimizing submodular functions. J. ACM 48(4), 761–777 (2001)

    MathSciNet  MATH  Google Scholar 

  19. Jeavons, P., Cohen, D., Cooper, M.C.: Constraints, consistency and closure. Artif. Intell. 101(1–2), 251–265 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  20. Jeavons, P.G., Cohen, D.A., Gyssens, M.: Closure properties of constraints. J. ACM 44(4), 527–548 (1997)

    MathSciNet  MATH  Google Scholar 

  21. Jeavons, P.G., Cooper, M.C.: Tractable constraints on ordered domains. Artif. Intell. 79(2), 327–339 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  22. Jeavons, P.G., Živný, S.: Tractable valued constraints. In: Bordeaux, L., Hamadi, Y., Kohli, P., Mateescu, R. (eds.) Tractability: Practical Approaches to Hard Problems. Cambridge University Press, Cambridge (2012, to appear)

    Google Scholar 

  23. Jonsson, P., Kuivinen, F., Thapper, J.: Min CSP on four elements: moving beyond submodularity. In: Proceedings of the 17th International Conference on Principles and Practice of Constraint Programming (CP’11). Lecture Notes in Computer Science, vol. 6876, pp. 438–453. Springer, Berlin (2011)

    Google Scholar 

  24. Khot, S.: On the unique games conjecture. In: Proceedings of the 25th Annual IEEE Conference on Computational Complexity (CCC’10), pp. 99–121. IEEE Computer Society, Los Alamitos (2010). Invited survey

    Chapter  Google Scholar 

  25. Kolmogorov, V.: Submodularity on a tree: unifying l ♯-convex and bisubmodular functions. In: Proceedings of the 36th International Symposium on Mathematical Foundations of Computer Science (MFCS’11). Lecture Notes in Computer Science, vol. 6907, pp. 400–411. Springer, Berlin (2011)

    Google Scholar 

  26. Kolmogorov, V.: Minimizing a sum of submodular functions. Discrete Appl. Math. 160(15), 2246–2258 (2012)

    Article  MathSciNet  Google Scholar 

  27. Kolmogorov, V., Živný, S.: The complexity of conservative valued CSPs. In: Proceedings of the 23rd Annual ACM-SIAM Symposium on Discrete Algorithms (SODA’12), pp. 750–759. SIAM, Philadelphia (2012). Full version available on arXiv:1110.2809

    Google Scholar 

  28. Krokhin, A., Larose, B.: Maximizing supermodular functions on product lattices, with application to maximum constraint satisfaction. SIAM J. Discrete Math. 22(1), 312–328 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  29. Kuivinen, F.: On the complexity of submodular function minimisation on diamonds. Discrete Optim. 8(3), 459–477 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  30. McCormick, S.T., Fujishige, S.: Strongly polynomial and fully combinatorial algorithms for bisubmodular function minimization. Math. Program. 122(1), 87–120 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  31. Qi, L.: Directed submodularity, ditroids and directed submodular flows. Math. Program. 42(1–3), 579–599 (1988)

    Article  MATH  Google Scholar 

  32. Raghavendra, P.: Approximating NP-hard problems: efficient algorithms and their limits. Ph.D. thesis, University of Washington (2009)

    Google Scholar 

  33. Rossi, F., van Beek, P., Walsh, T. (eds.): The Handbook of Constraint Programming. Elsevier, Amsterdam (2006)

    Google Scholar 

  34. Schrijver, A.: A combinatorial algorithm minimizing submodular functions in strongly polynomial time. J. Comb. Theory, Ser. B 80(2), 346–355 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  35. Thapper, J., Živný, S.: The power of linear programming for valued CSPs. In: Proceedings of the 53rd Annual IEEE Symposium on Foundations of Computer Science (FOCS’12). IEEE, New York (2012). Preliminary version available on arXiv:1204.1079

    Google Scholar 

  36. Živný, S.: The complexity and expressive power of valued constraints. Ph.D. thesis, University of Oxford (2009)

    Google Scholar 

  37. Živný, S., Cohen, D.A., Jeavons, P.G.: The expressive power of binary submodular functions. Discrete Appl. Math. 157(15), 3347–3358 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  38. Živný, S., Jeavons, P.G.: Classes of submodular constraints expressible by graph cuts. Constraints 15(3), 430–452 (2010)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. University College, University of Oxford, Oxford, UK

    Stanislav Živný

Authors
  1. Stanislav Živný
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 2012 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Živný, S. (2012). Tractable Languages. In: The Complexity of Valued Constraint Satisfaction Problems. Cognitive Technologies. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-33974-5_6

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-33974-5_6

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-33973-8

  • Online ISBN: 978-3-642-33974-5

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation