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.
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Notes
- 1.
The set of short supports depends on the domains \(D_s\). We always use the initial domains. Elsewhere, short supports are generated using the current domains D but these sets are not necessarily short supports after backtracking [18, 19]. A support of either type is valid iff all literals in it are valid.
References
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
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
Bacchus, F.: GAC via unit propagation. In: Bessière, C. (ed.) CP 2007. LNCS, vol. 4741, pp. 133–147. Springer, Heidelberg (2007)
Biere, A.: Lingeling, plingeling and treengeling entering the sat competition 2013. In: Proceedings of SAT Competition, pp. 51–52 (2013)
Biere, A., Heule, M., van Maaren, H.: Handbook of Satisfiability, vol. 185. IOS Press, Amsterdam (2009)
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
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)
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
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)
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)
Jefferson, C., Moore, N., Nightingale, P., Petrie, K.E.: Implementing logical connectives in constraint programming. Artif. Intell. 174, 1407–1429 (2010)
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)
Katsirelos, G., Walsh, T.: A compression algorithm for large arity extensional constraints. In: Proceedings of the CP 2007, pp. 379–393 (2007)
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)
Marques-Silva, J.: Practical applications of boolean satisfiability. In: 9th International Workshop on Discrete Event Systems (WODES 2008), pp. 74–80 (2008)
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)
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)
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)
Nightingale, P., Gent, I.P., Jefferson, C., Miguel, I.: Short and long supports for constraint propagation. J. Artif. Intell. Res. 46, 1–45 (2013)
Nightingale, P., Rendl, A.: Essence’ description (2016). arXiv:1601.02865 [cs.AI]
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)
Régin, J.: Improving the expressiveness of table constraints. In: The 10th International Workshop on Constraint Modelling and Reformulation (ModRef 2011) (2011)
Shang, Y., Wah, B.W.: A discrete Lagrangian-based global-search method for solving satisfiability problems. J. Glob. Optimi. 12(1), 61–99 (1998)
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
Simonis, H., O’Sullivan, B.: Search strategies for rectangle packing. In: Proceedings of the CP 2008, pp. 52–66 (2008)
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)
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.
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Akgün, Ö., Gent, I.P., Jefferson, C., Miguel, I., Nightingale, P. (2016). Exploiting Short Supports for Improved Encoding of Arbitrary Constraints into SAT. In: Rueher, M. (eds) Principles and Practice of Constraint Programming. CP 2016. Lecture Notes in Computer Science(), vol 9892. Springer, Cham. https://doi.org/10.1007/978-3-319-44953-1_1
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