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On Minimal Weighted Clones

  • Conference paper
Principles and Practice of Constraint Programming – CP 2011 (CP 2011)

Part of the book series: Lecture Notes in Computer Science ((LNPSE,volume 6876))

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Abstract

The connection between the complexity of constraint languages and clone theory, discovered by Cohen and Jeavons in a series of papers, has been a fruitful line of research on the complexity of CSPs. In a recent result, Cohen et al. [14] have established a Galois connection between the complexity of valued constraint languages and so-called weighted clones. In this paper, we initiate the study of weighted clones. Firstly, we prove an analogue of Rosenberg’s classification of minimal clones for weighted clones. Secondly, we show minimality of several weighted clones whose support clone is generated by a single minimal operation. Finally, we classify all Boolean weighted clones. This classification implies a complexity classification of Boolean valued constraint languages obtained by Cohen et al. [13]

This research was supported by EPSRC grant EP/F01161X/1. Stanislav Živný is supported by a Junior Research Fellowship at University College, Oxford.

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References

  1. Barto, L., Kozik, M., Maróti, M., Niven, T.: CSP dichotomy for special triads. Proceedings of the American Mathematical Society 137(9), 2921–2934 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  2. Barto, L., Kozik, M., Niven, T.: The CSP dichotomy holds for digraphs with no sources and no sinks. SIAM Journal on Computing 38(5), 1782–1802 (2009)

    Article  MATH  Google Scholar 

  3. Barto, L.: The dichotomy for conservative constraint satisfaction problems revisited. In: Proc. of LICS 2011 (2011)

    Google Scholar 

  4. Barto, L., Kozik, M.: Constraint Satisfaction Problems of Bounded Width. In: Proc. of FOCS 2009, pp. 461–471 (2009)

    Google Scholar 

  5. Berman, J., Idziak, P., Marković, P., McKenzie, R., Valeriote, M., Willard, R.: Varieties with few subalgebras of powers. Trans. of AMS 362(3), 1445–1473 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bulatov, A., Krokhin, A., Jeavons, P.: The complexity of maximal constraint languages. In: Proc. of STOC 2001, pp. 667–674 (2001)

    Google Scholar 

  7. Bulatov, A.: A Graph of a Relational Structure and Constraint Satisfaction Problems. In: Proc. of LICS 2004, pp. 448–457 (2004)

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  9. Bulatov, A., Krokhin, A., Jeavons, P.: Classifying the Complexity of Constraints using Finite Algebras. SIAM Journal on Computing 34(3), 720–742 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  10. Bulatov, A.A.: Tractable Conservative Constraint Satisfaction Problems. In: Proc. of LICS 2003, pp. 321–330 (2003)

    Google Scholar 

  11. Cohen, D.A., Cooper, M.C., Jeavons, P.G.: An Algebraic Characterisation of Complexity for Valued Constraints. In: Benhamou, F. (ed.) CP 2006. LNCS, vol. 4204, pp. 107–121. Springer, Heidelberg (2006)

    Chapter  Google Scholar 

  12. Cohen, D.A., Cooper, M.C., Jeavons, P.G.: Generalising submodularity and Horn clauses: Tractable optimization problems defined by tournament pair multimorphisms. Theoretical Computer Science 401(1-3), 36–51 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  13. Cohen, D.A., Cooper, M.C., Jeavons, P.G., Krokhin, A.A.: The Complexity of Soft Constraint Satisfaction. Artificial Intelligence 170(11), 983–1016 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  14. Cohen, D., Creed, P., Jeavons, P., Živný, S.: An algebraic theory of complexity for valued constraints: Establishing a Galois connection. In: Murlak, F., Sankowski, P. (eds.) Mathematical Foundations of Computer Science 2011. LNCS, vol. 6907, pp. 231–242. Springer, Heidelberg (2011)

    Chapter  Google Scholar 

  15. Creignou, N.: A dichotomy theorem for maximum generalized satisfiability problems. Journal of Computer and System Sciences 51(3), 511–522 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  16. Creignou, N., Khanna, S., Sudan, M.: Complexity Classification of Boolean Constraint Satisfaction Problems. SIAM Monographs on Discrete Mathematics and Applications, vol. 7. SIAM, Philadelphia (2001)

    Book  MATH  Google Scholar 

  17. Csákány, B.: Minimal clones – a minicourse. Algebra Universalis 54(1), 73–89 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  18. Denecke, K., Wismath, S.: Universal Algebra and Applications in Theoretical Computer Science. Chapman and Hall/CRC Press (2002)

    Google Scholar 

  19. Feder, T., Vardi, M.: The Computational Structure of Monotone Monadic SNP and Constraint Satisfaction: A Study through Datalog and Group Theory. SIAM Journal on Computing 28(1), 57–104 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  20. Jeavons, P.: On the Algebraic Structure of Combinatorial Problems. Theoretical Computer Science 200(1-2), 185–204 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  21. Jeavons, P., Cohen, D., Gyssens, M.: Closure Properties of Constraints. Journal of the ACM 44(4), 527–548 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  22. Kolmogorov, V., Živný, S.: The complexity of conservative finite-valued CSPs. Technical repport arXiv:1008.1555 (August 2010)

    Google Scholar 

  23. Kolmogorov, V., Živný, S.: The complexity of conservative valued CSPs (submitted for publication, 2011)

    Google Scholar 

  24. Nemhauser, G., Wolsey, L.: Integer and Combinatorial Optimization (1988)

    Google Scholar 

  25. Post, E.: The two-valued iterative systems of mathematical logic. Annals of Mathematical Studies, vol. 5. Princeton University Press, Princeton (1941)

    MATH  Google Scholar 

  26. Rosenberg, I.: Minimal Clones I: the five types. In: Lectures in Universal Algebra (Proc. Conf. Szeged 1983). Colloq. Math. Soc. Janos Bolyai, vol. 43, pp. 405–427. North-Holland, Amsterdam (1986)

    Google Scholar 

  27. Schaefer, T.: The Complexity of Satisfiability Problems. In: Proc. of STOC 1978, pp. 216–226 (1978)

    Google Scholar 

  28. Schrijver, A.: Theory of linear and integer programming (1986)

    Google Scholar 

  29. Schrijver, A.: Combinatorial Optimization: Polyhedra and Efficiency (2003)

    Google Scholar 

  30. Świerczkowski, S.: Algebras which are independently generated by every n elements. Fundamenta Mathematicae 49, 93–104 (1960)

    MathSciNet  MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Computer Science, Royal Holloway, University of London, UK

    Páidí Creed

  2. University College, University of Oxford, UK

    Stanislav Živný

Authors
  1. Páidí Creed
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  2. Stanislav Živný
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Editors and Affiliations

  1. Department of Computer Science and Engineering, The Chinese University of Hong Kong, Shatin, N.T., Hong Kong, China

    Jimmy Lee

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Creed, P., Živný, S. (2011). On Minimal Weighted Clones. In: Lee, J. (eds) Principles and Practice of Constraint Programming – CP 2011. CP 2011. Lecture Notes in Computer Science, vol 6876. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-23786-7_18

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  • DOI: https://doi.org/10.1007/978-3-642-23786-7_18

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-23785-0

  • Online ISBN: 978-3-642-23786-7

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