Constraints

, Volume 21, Issue 4, pp 463–494 | Cite as

Extending SMT solvers with support for finite domain alldifferent constraint

Article

Abstract

In this paper we consider integration of SMT solvers with the filtering algorithms for the finite domain alldifferent constraint. Such integration makes SMT solvers suitable for solving constraint satisfaction problems with the alldifferent constraint involved. First, we present a novel algorithm for explaining inconsistencies and propagations in the alldifferent constraint. We compare it to Katsirelos’ algorithm and flow-based algorithms that are commonly used for that purpose. Then we describe our DPLL(T)-compliant SMT theory solver for constraint satisfaction problems that include alldifferent constraints. We also provide an experimental evaluation of our approach.

Keywords

SMT solving CSP solving alldifferent constraint Explanation algorithms 

References

  1. 1.
    Bankovic, M., & Maric, F. (2010). An Alldifferent constraint solver in SMT. In 8th International Workshop on Satisfiability Modulo Theories.Google Scholar
  2. 2.
    Barrett, C., Sebastiani, R., Seshia, S.A., & Tinelli, C. (2009). Satisfiability Modulo Theories. In Handbook of Satisfiability, chap. 26, pp. 825–885. IOS Press.Google Scholar
  3. 3.
    Barrett, C., Stump, A., & Tinelli, C. (2010). The SMT-LIB Standard: Version 2.0, http://smtlib.cs.uiowa.edu/papers/smt-lib-reference-v2.0-r12.09.09.pdf.
  4. 4.
    Berge, C. (1970). Graphes et hypergraphes.Google Scholar
  5. 5.
    Bofill, M., Palahí, M., Suy, J., & Villaret, M. (2012). Solving constraint satisfaction problems with SAT modulo theories. Constraints, 17(3), 273–303.MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Bozzano, M., Bruttomesso, R., Cimatti, A., Junttila, T., Van Rossum, P., Schulz, S., & Sebastiani, R. (2005). An incremental and layered procedure for the satisfiability of linear arithmetic logic. In Tools and Algorithms for the Construction and Analysis of Systems, pp. 317–333. Springer.Google Scholar
  7. 7.
    Davis, M., Logemann, G., & Loveland, D. (1962). A machine program for theorem-proving. Communications of the ACM, 5 (7), 394–397. doi:10.1145/368273.368557.MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Downing, N., Feydy, T., & Stuckey, P.J. (2012). Explaining alldifferent. In Proceedings of the Thirty-fifth Australasian Computer Science Conference-Volume 122, pp. 115–124. Australian Computer Society, Inc.Google Scholar
  9. 9.
    Downing, N., Feydy, T., & Stuckey, P.J. (2012). Explaining flow-based propagation. In Integration of AI and OR Techniques in Contraint Programming for Combinatorial Optimzation Problems, pp. 146–162. Springer.Google Scholar
  10. 10.
    Dutertre, B., & De Moura, L. (2006). The Yices SMT solver. Tool paper at http://yices.csl.sri.com/tool-paper.pdf2, Vol. 2.
  11. 11.
    Ford, L.R., & Fulkerson, D.R. (1956). Maximal flow through a network. Canadian Journal Mathematics, 8(3), 399–404.MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Ganzinger, H., Hagen, G., Nieuwenhuis, R., Oliveras, A., & Tinelli, C. (2004). DPLL(T): Fast Decision Procedures. In CAV, Lecture Notes in Computer Science, vol. 3114, pp. 175–188. Springer.Google Scholar
  13. 13.
    Gent, I.P., Jefferson, C., & Miguel, I. (2006). Minion: A fast scalable constraint solver. In ECAI, vol. 141, pp. 98–102.Google Scholar
  14. 14.
    Gent, I.P., & Lynce, I. (2005). A SAT encoding for the social golfer problem. Modelling and Solving Problems with Constraints, 2.Google Scholar
  15. 15.
    Gent, I.P., Miguel, I., & Moore, N.C. (2010). Lazy explanations for constraint propagators. In Practical Aspects of Declarative Languages, pp. 217–233. Springer.Google Scholar
  16. 16.
    Gent, I.P., Miguel, I., & Nightingale, P. (2008). Generalised arc consistency for the alldifferent constraint: An empirical survey. Artificial Intelligence, 172(18), 1973–2000.MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Gomes, C., & Shmoys, D. (2002). Completing quasigroups or latin squares: A structured graph coloring problem. In proceedings of the Computational Symposium on Graph Coloring and Generalizations, pp. 22–39.Google Scholar
  18. 18.
    van Hoeve, W.J. (2001). The alldifferent constraint: A survey. arXiv:preprintcs/0105015.
  19. 19.
    Janicic, P. (2010). Uniform Reduction to SAT Vol. 8.Google Scholar
  20. 20.
    Janicic, P. (2010). Maric, F., Uniform reduction to SMT.Google Scholar
  21. 21.
    Katsirelos, G. (2008). Nogood processing in CSPs. Ph.D. thesis: University of Toronto.Google Scholar
  22. 22.
    Krstic, S., & Goel, A. (2007). Architecting Solvers for SAT Modulo Theories: Nelson-Oppen with DPLL. In FroCoS, Lecture Notes in Computer Science, vol. 4720, pp. 1–27. Springer.Google Scholar
  23. 23.
    Lewis, R. (2007). Metaheuristics can solve sudoku puzzles. Journal of Heuristics, 13(4), 387–401.CrossRefGoogle Scholar
  24. 24.
    Marques-Silva, J., Lynce, I., & Malik, S. (2009). Conflict-Driven Clause Learning SAT Solvers. In Handbook of Satisfiability, chap. 4, pp. 131–155. IOS Press.Google Scholar
  25. 25.
    Moore, N. (2011). Improving the efficiency of learning CSP solvers. Ph.D. thesis: University of St Andrews.Google Scholar
  26. 26.
    Nieuwenhuis, R. (2009). Sat modulo theories: Enhancing SAT with special-purpose algorithms. In Theory and Applications of Satisfiability Testing-SAT 2009, pp. 1–1. Springer.Google Scholar
  27. 27.
    Nieuwenhuis, R., & Oliveras, A. (2007). Rodríguez-Carbonell, E., Rubio, A.: Challenges in satisfiability modulo theories. In Term Rewriting and Applications, pp. 2–18. Springer.Google Scholar
  28. 28.
    Ohrimenko, O., Stuckey, P.J., & Codish, M. (2009). Propagation via lazy clause generation. Constraints, 14(3), 357–391.MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Petke, J., & Jeavons, P. (2011). The order encoding: from tractable CSP to tractable SAT. In Theory and Applications of Satisfiability Testing-SAT 2011, pp. 371–372. Springer.Google Scholar
  30. 30.
    Régin, J.C. (1994). A filtering algorithm for constraints of difference in CSPs. In AAAI, vol. 94, pp. 362–367.Google Scholar
  31. 31.
    Régin, J.C. (1996). Generalized arc consistency for global cardinality constraint. In Proceedings of the thirteenth national conference on Artificial intelligence-Volume 1, pp. 209–215. AAAI Press.Google Scholar
  32. 32.
    Rochart, G., Jussien, N., & Laburthe, F. (2003). Challenging explanations for global constraints. In CP03 Workshop on User-Interaction in Constraint Satisfaction (UICS03), pp. 31–43.Google Scholar
  33. 33.
    Rossi, F., Van Beek, P., & Walsh, T. (2006). Handbook of constraint programming: Elsevier.Google Scholar
  34. 34.
    Schulte, C., & Stuckey, P.J. (2004). Speeding up constraint propagation. In Principles and Practice of Constraint Programming–CP 2004, pp. 619–633. Springer.Google Scholar
  35. 35.
    Schulte, C., & Stuckey, P.J. (2005). When do bounds and domain propagation lead to the same search space. ACM Transactions on Programming Languages and Systems (TOPLAS), 27(3), 388–425.CrossRefGoogle Scholar
  36. 36.
    Sellmann, M., & Kadioglu, S. (2008). Dichotomic search protocols for constrained optimization. In Principles and Practice of Constraint Programming, pp. 251–265. Springer.Google Scholar
  37. 37.
    Simonis, H. (2008). Kakuro as a constraint problem. In Proc. seventh Int. Works on Constraint Modelling and Reformulation.Google Scholar
  38. 38.
    Stojadinović, M., & Marić, F. (2014). meSAT: multiple encodings of CSP to SAT. Constraints, 1–24.Google Scholar
  39. 39.
    Stuckey, P.J. (2010). Lazy clause generation: Combining the power of SAT and CP (and MIP?) solving. In Integration of AI and OR Techniques in Constraint Programming for Combinatorial Optimization Problems, pp. 5–9. Springer.Google Scholar
  40. 40.
    Tamura, N., Taga, A., Kitagawa, S., & Banbara, M. (2009). Compiling finite linear CSP into SAT. Constraints, 14(2), 254–272.MathSciNetCrossRefMATHGoogle Scholar
  41. 41.
    Tarjan, R. (1972). Depth-first search and linear graph algorithms. SIAM Journal Computing, 1(2), 146–160.MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Faculty of MathematicsUniversity of BelgradeBelgradeSerbia

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