Advertisement

Exploiting Short Supports for Improved Encoding of Arbitrary Constraints into SAT

  • Özgür Akgün
  • Ian P. Gent
  • Christopher Jefferson
  • Ian Miguel
  • Peter  NightingaleEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9892)

Abstract

Encoding to SAT and applying a highly efficient modern SAT solver is an increasingly popular method of solving finite-domain constraint problems. In this paper we study encodings of arbitrary constraints where unit propagation on the encoding provides strong reasoning. Specifically, unit propagation on the encoding simulates generalised arc consistency on the original constraint. To create compact and efficient encodings we use the concept of short support. Short support has been successfully applied to create efficient propagation algorithms for arbitrary constraints. A short support of a constraint is similar to a satisfying tuple however a short support is not required to assign every variable in scope. Some variables are left free to take any value. In some cases a short support representation is smaller than the table of satisfying tuples by an exponential factor. We present two encodings based on short supports and evaluate them on a set of benchmark problems, demonstrating a substantial improvement over the state of the art.

Notes

Acknowledgements

We would like to thank the EPSRC for funding this work through grants EP/H004092/1, EP/K015745/1, and EP/M003728/1. In addition, Dr Jefferson is funded by a Royal Society University Research Fellowship.

References

  1. 1.
    Abío, I., Mayer-Eichberger, V., Stuckey, P.J.: Encoding linear constraints with implication chains to CNF. In: Pesant, G. (ed.) CP 2015. LNCS, vol. 9255, pp. 3–11. Springer, Heidelberg (2015). doi: 10.1007/978-3-319-23219-5_1 Google Scholar
  2. 2.
    Asín, R., Nieuwenhuis, R., Oliveras, A., Rodríguez-Carbonell, E.: Cardinality networks: a theoretical and empirical study. Constraints 16(2), 195–221 (2011). doi: 10.1007/s10601-010-9105-0 MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bacchus, F.: GAC via unit propagation. In: Bessière, C. (ed.) CP 2007. LNCS, vol. 4741, pp. 133–147. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  4. 4.
    Biere, A.: Lingeling, plingeling and treengeling entering the sat competition 2013. In: Proceedings of SAT Competition, pp. 51–52 (2013)Google Scholar
  5. 5.
    Biere, A., Heule, M., van Maaren, H.: Handbook of Satisfiability, vol. 185. IOS Press, Amsterdam (2009)zbMATHGoogle Scholar
  6. 6.
    Brain, M., Hadarean, L., Kroening, D., Martins, R.: Automatic generation of propagation complete SAT encodings. In: Jobstmann, B., Leino, K.R.M. (eds.) VMCAI 2016. LNCS, vol. 9583, pp. 536–556. Springer, Heidelberg (2016). doi: 10.1007/978-3-662-49122-5_26 CrossRefGoogle Scholar
  7. 7.
    Cheng, K.C.K., Yap, R.H.C.: An MDD-based generalized arc consistency algorithm for positive and negative table constraints and some global constraints. Constraints 15(2), 265–304 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Eén, N., Sörensson, N.: Translating pseudo-boolean constraints into SAT. JSAT 2(1–4), 1–26 (2006). http://jsat.ewi.tudelft.nl/content/volume2/JSAT2_1_Een.pdf zbMATHGoogle Scholar
  9. 9.
    Gent, I.P., Jefferson, C., Miguel, I., Nightingale, P.: Generating special-purpose stateless propagators for arbitrary constraints. In: Cohen, D. (ed.) CP 2010. LNCS, vol. 6308, pp. 206–220. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  10. 10.
    Gent, I.P., Nightingale, P.: A new encoding of alldifferent into SAT. In: Proceedings 3rd International Workshop on Modelling and Reformulating Constraint Satisfaction Problems (ModRef 2004), pp. 95–110 (2004)Google Scholar
  11. 11.
    Jefferson, C., Moore, N., Nightingale, P., Petrie, K.E.: Implementing logical connectives in constraint programming. Artif. Intell. 174, 1407–1429 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Jefferson, C., Nightingale, P.: Extending simple tabular reduction with short supports. In: Proceedings of 23rd International Joint Conference on Artificial Intelligence (IJCAI), pp. 573–579 (2013)Google Scholar
  13. 13.
    Katsirelos, G., Walsh, T.: A compression algorithm for large arity extensional constraints. In: Proceedings of the CP 2007, pp. 379–393 (2007)Google Scholar
  14. 14.
    Mairy, J.-B., Deville, Y., Lecoutre, C.: The smart table constraint. In: Michel, L. (ed.) CPAIOR 2015. LNCS, vol. 9075, pp. 271–287. Springer, Heidelberg (2015)Google Scholar
  15. 15.
    Marques-Silva, J.: Practical applications of boolean satisfiability. In: 9th International Workshop on Discrete Event Systems (WODES 2008), pp. 74–80 (2008)Google Scholar
  16. 16.
    Moskewicz, M.W., Madigan, C.F., Zhao, Y., Zhang, L., Malik, S.: Chaff: Engineering an efficient SAT solver. In: Proceedings of the 38th Annual Design Automation Conference, pp. 530–535. ACM (2001)Google Scholar
  17. 17.
    Nightingale, P., Akgün, Ö., Gent, I.P., Jefferson, C., Miguel, I.: Automatically improving constraint models in savile row through associative-commutative common subexpression elimination. In: O’Sullivan, B. (ed.) CP 2014. LNCS, vol. 8656, pp. 590–605. Springer, Heidelberg (2014)Google Scholar
  18. 18.
    Nightingale, P., Gent, I.P., Jefferson, C., Miguel, I.: Exploiting short supports for generalised arc consistency for arbitrary constraints. In: Proceedings of the IJCAI 2011, pp. 623–628 (2011)Google Scholar
  19. 19.
    Nightingale, P., Gent, I.P., Jefferson, C., Miguel, I.: Short and long supports for constraint propagation. J. Artif. Intell. Res. 46, 1–45 (2013)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Nightingale, P., Rendl, A.: Essence’ description (2016). arXiv:1601.02865 [cs.AI]
  21. 21.
    Nightingale, P., Spracklen, P., Miguel, I.: Automatically improving SAT encoding of constraint problems through common subexpression elimination in savile row. In: Pesant, G. (ed.) CP 2015. LNCS, vol. 9255, pp. 330–340. Springer, Heidelberg (2015)Google Scholar
  22. 22.
    Régin, J.: Improving the expressiveness of table constraints. In: The 10th International Workshop on Constraint Modelling and Reformulation (ModRef 2011) (2011)Google Scholar
  23. 23.
    Shang, Y., Wah, B.W.: A discrete Lagrangian-based global-search method for solving satisfiability problems. J. Glob. Optimi. 12(1), 61–99 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Marques-Silva, J., Lynce, I.: Towards robust CNF encodings of cardinality constraints. In: Bessière, C. (ed.) CP 2007. LNCS, vol. 4741, pp. 483–497. Springer, Heidelberg (2007). doi: 10.1007/978-3-540-74970-7_35 CrossRefGoogle Scholar
  25. 25.
    Simonis, H., O’Sullivan, B.: Search strategies for rectangle packing. In: Proceedings of the CP 2008, pp. 52–66 (2008)Google Scholar
  26. 26.
    Stuckey, P.J., Tack, G.: MiniZinc with functions. In: Gomes, C., Sellmann, M. (eds.) CPAIOR 2013. LNCS, vol. 7874, pp. 268–283. Springer, Heidelberg (2013)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  • Özgür Akgün
    • 1
  • Ian P. Gent
    • 1
  • Christopher Jefferson
    • 1
  • Ian Miguel
    • 1
  • Peter  Nightingale
    • 1
    Email author
  1. 1.School of Computer ScienceUniversity of St AndrewsSt AndrewsUK

Personalised recommendations