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Incremental Cardinality Constraints for MaxSAT

  • Ruben Martins
  • Saurabh Joshi
  • Vasco Manquinho
  • Inês Lynce
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8656)

Abstract

Maximum Satisfiability (MaxSAT) is an optimization variant of the Boolean Satisfiability (SAT) problem. In general, MaxSAT algorithms perform a succession of SAT solver calls to reach an optimum solution making extensive use of cardinality constraints. Many of these algorithms are non-incremental in nature, i.e. at each iteration the formula is rebuilt and no knowledge is reused from one iteration to another. In this paper, we exploit the knowledge acquired across iterations using novel schemes to use cardinality constraints in an incremental fashion. We integrate these schemes with several MaxSAT algorithms. Our experimental results show a significant performance boost for these algorithms as compared to their non-incremental counterparts. These results suggest that incremental cardinality constraints could be beneficial for other constraint solving domains.

Keywords

Conjunctive Normal Form Cardinality Constraint Unit Clause Conjunctive Normal Form Formula Relaxation Variable 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Abío, I., Stuckey, P.J.: Conflict Directed Lazy Decomposition. In: Milano, M. (ed.) CP 2012. LNCS, vol. 7514, pp. 70–85. Springer, Heidelberg (2012)Google Scholar
  2. 2.
    Ansótegui, C., Bonet, M.L., Gabàs, J., Levy, J.: Improving WPM2 for (Weighted) Partial MaxSAT. In: Schulte, C. (ed.) CP 2013. LNCS, vol. 8124, pp. 117–132. Springer, Heidelberg (2013)Google Scholar
  3. 3.
    Ansótegui, C., Bonet, M.L., Levy, J.: Solving (Weighted) Partial MaxSAT through Satisfiability Testing. In: Kullmann (ed.) [30], pp. 427–440Google Scholar
  4. 4.
    Ansótegui, C., Bonet, M.L., Levy, J.: A New Algorithm for Weighted Partial MaxSAT. In: Fox, M., Poole, D. (eds.) AAAI Conference on Artificial Intelligence. AAAI Press (2010)Google Scholar
  5. 5.
    Argelich, J., Berre, D.L., Lynce, I., Marques-Silva, J., Rapicault, P.: Solving Linux Upgradeability Problems Using Boolean Optimization. In: Workshop on Logics for Component Configuration, pp. 11–22 (2010)Google Scholar
  6. 6.
    Asín, R., Nieuwenhuis, R.: Curriculum-based course timetabling with SAT and MaxSAT. Annals of Operations Research, 1–21 (2012)Google Scholar
  7. 7.
    Asín, R., Nieuwenhuis, R., Oliveras, A., Rodríguez-Carbonell, E.: Cardinality Networks: a theoretical and empirical study. Constraints 16(2), 195–221 (2011)zbMATHMathSciNetGoogle Scholar
  8. 8.
    Audemard, G., Lagniez, J.M., Simon, L.: Improving Glucose for Incremental SAT Solving with Assumptions: Application to MUS Extraction. In: Järvisalo, M., Van Gelder, A. (eds.) SAT 2013. LNCS, vol. 7962, pp. 309–317. Springer, Heidelberg (2013)Google Scholar
  9. 9.
    Bailleux, O., Boufkhad, Y.: Efficient CNF Encoding of Boolean Cardinality Constraints. In: Rossi, F. (ed.) CP 2003. LNCS, vol. 2833, pp. 108–122. Springer, Heidelberg (2003)Google Scholar
  10. 10.
    Barrett, C., Stump, A., Tinelli, C.: The SMT-LIB Standard: Version 2.0. Tech. rep., Department of Computer Science, The University of Iowa (2010), www.SMT-LIB.org
  11. 11.
    Büttner, M., Rintanen, J.: Satisfiability Planning with Constraints on the Number of Actions. In: Biundo, S., Myers, K.L., Rajan, K. (eds.) International Conference on Automated Planning and Scheduling, pp. 292–299 (2005)Google Scholar
  12. 12.
    Chen, Y., Safarpour, S., Marques-Silva, J., Veneris, A.G.: Automated Design Debugging With Maximum Satisfiability. IEEE Transactions on CAD of Integrated Circuits and Systems 29(11), 1804–1817 (2010)Google Scholar
  13. 13.
    Cheng, K.C.K., Yap, R.H.C.: Maintaining Generalized Arc Consistency on Ad-Hoc n-Ary Boolean Constraints. In: Brewka, G., Coradeschi, S., Perini, A., Traverso, P. (eds.) European Conference on Artificial Intelligence. Frontiers in Artificial Intelligence and Applications, vol. 141, pp. 78–82. IOS Press (2006)Google Scholar
  14. 14.
    Cimatti, A., Sebastiani, R. (eds.): SAT 2012. LNCS, vol. 7317, pp. 2012–2015. Springer, Heidelberg (2012)zbMATHGoogle Scholar
  15. 15.
    Davies, J., Bacchus, F.: Exploiting the Power of mip Solvers in maxsat. In: Järvisalo, M., Van Gelder, A. (eds.) SAT 2013. LNCS, vol. 7962, pp. 166–181. Springer, Heidelberg (2013)Google Scholar
  16. 16.
    Davies, J., Bacchus, F.: Postponing Optimization to Speed Up MAXSAT Solving. In: Schulte, C. (ed.) CP 2013. LNCS, vol. 8124, pp. 247–262. Springer, Heidelberg (2013)Google Scholar
  17. 17.
    Debruyne, R.: Arc-Consistency in Dynamic CSPs Is No More Prohibitive. In: International Conference on Tools with Artificial Intelligence, pp. 299–307. IEEE (1996)Google Scholar
  18. 18.
    Dechter, R., Dechter, A.: Belief Maintenance in Dynamic Constraint Networks. In: Shrobe, H.E., Mitchell, T.M., Smith, R.G. (eds.) AAAI Conference on Artificial Intelligence, pp. 37–42. AAAI Press / The MIT Press (1988)Google Scholar
  19. 19.
    Dutertre, B., de Moura, L.M.: A Fast Linear-Arithmetic Solver for DPLL(T). In: Ball, T., Jones, R.B. (eds.) CAV 2006. LNCS, vol. 4144, pp. 81–94. Springer, Heidelberg (2006)Google Scholar
  20. 20.
    Eén, N., Sörensson, N.: Translating Pseudo-Boolean Constraints into SAT. Journal on Satisfiability, Boolean Modeling and Computation 2, 1–26 (2006)zbMATHGoogle Scholar
  21. 21.
    Eén, N., Sörensson, N.: An Extensible SAT-solver. In: Giunchiglia, E., Tacchella, A. (eds.) SAT 2003. LNCS, vol. 2919, pp. 502–518. Springer, Heidelberg (2004)Google Scholar
  22. 22.
    Eén, N., Sörensson, N.: Temporal induction by incremental SAT solving. Electronic Notes in Theoretical Computer Science 89(4), 543–560 (2003)Google Scholar
  23. 23.
    Fu, Z., Malik, S.: On Solving the Partial MAX-SAT Problem. In: Biere, A., Gomes, C.P. (eds.) SAT 2006. LNCS, vol. 4121, pp. 252–265. Springer, Heidelberg (2006)Google Scholar
  24. 24.
    Graça, A., Lynce, I., Marques-Silva, J., Oliveira, A.L.: Efficient and Accurate Haplotype Inference by Combining Parsimony and Pedigree Information. In: Horimoto, K., Nakatsui, M., Popov, N. (eds.) ANB 2010. LNCS, vol. 6479, pp. 38–56. Springer, Heidelberg (2012)Google Scholar
  25. 25.
    Heras, F., Morgado, A., Marques-Silva, J.: Core-guided binary search algorithms for maximum satisfiability. In: Burgard, W., Roth, D. (eds.) AAAI Conference on Artificial Intelligence. AAAI Press (2011)Google Scholar
  26. 26.
    Hooker, J.N.: Solving the incremental satisfiability problem. Journal of Logic Programming 15(1&2), 177–186 (1993)zbMATHMathSciNetGoogle Scholar
  27. 27.
    Jose, M., Majumdar, R.: Cause clue clauses: error localization using maximum satisfiability. In: Hall, M.W., Padua, D.A. (eds.) Programming Language Design and Implementation, pp. 437–446. ACM (2011)Google Scholar
  28. 28.
    Kadioglu, S., Malitsky, Y., Sellmann, M.: Non-Model-Based Search Guidance for Set Partitioning Problems. In: Hoffmann, J., Selman, B. (eds.) AAAI Conference on Artificial Intelligence. AAAI Press (2012)Google Scholar
  29. 29.
    Koshimura, M., Zhang, T., Fujita, H., Hasegawa, R.: QMaxSAT: A Partial Max-SAT Solver. Journal on Satisfiability, Boolean Modeling and Computation 8, 95–100 (2012)MathSciNetGoogle Scholar
  30. 30.
    Kullmann, O. (ed.): SAT 2009. LNCS, vol. 5584. Springer, Heidelberg (2009)zbMATHGoogle Scholar
  31. 31.
    Lagerkvist, M.Z., Schulte, C.: Advisors for Incremental Propagation. In: Bessière, C. (ed.) CP 2007. LNCS, vol. 4741, pp. 409–422. Springer, Heidelberg (2007)Google Scholar
  32. 32.
    Le Berre, D., Parrain, A.: The Sat4j library, release 2.2. Journal on Satisfiability, Boolean Modeling and Computation 7(2-3), 59–66 (2010)Google Scholar
  33. 33.
    Li, C.M., Manyà, F.: MaxSAT, Hard and Soft Constraints. In: Handbook of Satisfiability, pp. 613–631. IOS Press (2009)Google Scholar
  34. 34.
    Liffiton, M.H., Sakallah, K.A.: Algorithms for Computing Minimal Unsatisfiable Subsets of Constraints. Journal Automated Reasoning 40(1), 1–33 (2008)zbMATHMathSciNetGoogle Scholar
  35. 35.
    Lonsing, F., Egly, U.: Incremental QBF Solving. Computing Research Repository - arXiv abs/1402.2410 (2014)Google Scholar
  36. 36.
    Mahajan, Y.S., Fu, Z., Malik, S.: Zchaff2004: An efficient sat solver. In: H. Hoos, H., Mitchell, D.G. (eds.) SAT 2004. LNCS, vol. 3542, pp. 360–375. Springer, Heidelberg (2005)Google Scholar
  37. 37.
    Manolios, P., Papavasileiou, V.: Pseudo-Boolean Solving by incremental translation to SAT. In: Bjesse, P., Slobodová, A. (eds.) International Conference on Formal Methods in Computer-Aided Design, pp. 41–45. FMCAD Inc. (2011)Google Scholar
  38. 38.
    Manquinho, V., Marques-Silva, J., Planes, J.: Algorithms for Weighted Boolean Optimization. In: Kullmann (ed.) [30], pp. 495–508Google Scholar
  39. 39.
    Marin, P., Miller, C., Lewis, M.D.T., Becker, B.: Verification of partial designs using incremental QBF solving. In: Rosenstiel, W., Thiele, L. (eds.) Design, Automation, and Test in Europe Conference, pp. 623–628. IEEE (2012)Google Scholar
  40. 40.
    Marques-Silva, J., Planes, J.: On using unsatisfiability for solving Maximum Satisfiability. Tech. rep., Computing Research Repository, abs/0712.0097 (2007)Google Scholar
  41. 41.
    Martins, R., Manquinho, V., Lynce, I.: Parallel Search for Maximum Satisfiability. AI Communications 25(2), 75–95 (2012)MathSciNetGoogle Scholar
  42. 42.
    Martins, R., Manquinho, V., Lynce, I.: Open-WBO: a Modular MaxSAT Solver. In: Sinz, C., Egly, U. (eds.) SAT 2014. LNCS, vol. 8561, pp. 438–445. Springer, Heidelberg (2014)Google Scholar
  43. 43.
    Morgado, A., Heras, F., Liffiton, M., Planes, J., Marques-Silva, J.: Iterative and core-guided MaxSAT solving: A survey and assessment. Constraints 18(4), 478–534 (2013)MathSciNetGoogle Scholar
  44. 44.
    Morgado, A., Heras, F., Marques-Silva, J.: Improvements to Core-Guided Binary Search for MaxSAT. In: Cimatti, Sebastiani (eds.) [14], pp. 284–297Google Scholar
  45. 45.
    Nadel, A., Ryvchin, V.: Efficient SAT Solving under Assumptions. In: Cimatti, Sebastiani (eds.) [14], pp. 242–255Google Scholar
  46. 46.
    Sinz, C.: Towards an Optimal CNF Encoding of Boolean Cardinality Constraints. In: van Beek, P. (ed.) CP 2005. LNCS, vol. 3709, pp. 827–831. Springer, Heidelberg (2005)Google Scholar
  47. 47.
    Shtrichman, O.: Pruning Techniques for the SAT-Based Bounded Model Checking Problem. In: Margaria, T., Melham, T.F. (eds.) CHARME 2001. LNCS, vol. 2144, pp. 58–70. Springer, Heidelberg (2001)Google Scholar
  48. 48.
    van Beek, P.: Backtracking Search Algorithms. In: Rossi, F., van Beek, P., Walsh, T. (eds.) Handbook of Constraint Programming, ch. 4. Elsevier (2006)Google Scholar
  49. 49.
    van Hoeve, W.J., Katriel, I.: Global constraints. In: Rossi, F., van Beek, P., Walsh, T. (eds.) Handbook of Constraint Programming, ch. 6. Elsevier (2006)Google Scholar
  50. 50.
    Whittemore, J., Kim, J., Sakallah, K.A.: SATIRE: A New Incremental Satisfiability Engine. In: Design Automation Conference, pp. 542–545. ACM (2001)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Ruben Martins
    • 1
  • Saurabh Joshi
    • 1
  • Vasco Manquinho
    • 2
  • Inês Lynce
    • 2
  1. 1.Department of Computer ScienceUniversity of OxfordUK
  2. 2.INESC-ID / Instituto Superior TécnicoUniversidade de LisboaPortugal

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