Skip to main content
SpringerLink
Log in

Solving quantified constraint satisfaction problems with value selection rules

  • Research Article
  • Published:
Frontiers of Computer Science Aims and scope Submit manuscript

Abstract

Solving a quantified constraint satisfaction problem (QCSP) is usually a hard task due to its computational complexity. Exact algorithms play an important role in solving this problem, among which backtrack algorithms are effective. In a backtrack algorithm, an important step is assigning a variable by a chosen value when exploiting a branch, and thus a good value selection rule may speed up greatly. In this paper, we propose two value selection rules for existentially and universally quantified variables, respectively, to avoid unnecessary searching. The rule for universally quantified variables is prior to trying failure values in previous branches, and the rule for existentially quantified variables selects the promising values first. Two rules are integrated into the state-of-the-art QCSP solver, i.e., QCSP-Solve, which is an exact solver based on backtracking. We perform a number of experiments to evaluate improvements brought by our rules. From computational results, we can conclude that the new value selection rules speed up the solver by 5 times on average and 30 times at most. We also show both rules perform well particularly on instances with existentially and universally quantified variables occurring alternatively.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Chu Y, Luo C, Cai S, You H. Empirical investigation of stochastic local search for maximum satisfiability. Frontiers of Computer Science, 2019, 13(1): 86–98

    Article  Google Scholar 

  2. Zhang Z, Fu Q, Zhang X, Liu Q. Reasoning and predicting POMDP planning complexity via covering numbers. Frontiers of Computer Science, 2016, 10(4): 726–740

    Article  Google Scholar 

  3. Gao J, Wang J, Yin M. Experimental analyses on phase transitions in compiling satisfiability problems. Science China Information Sciences, 2015, 58(3): 1–11

    Article  Google Scholar 

  4. Amilhastre J, Fargier H, Marquis P. Consistency restoration and explanations in dynamic CSPs—application to configuration. Artificial Intelligence, 2002, 135(1–2): 199–234

    Article  MathSciNet  Google Scholar 

  5. Ghallab M, Nau D, Traverso P. Automated Planning: Theory and Practice. Elsevier, 2004

  6. Palmieri A, Lallouet A. Constraint games revisited. In: Proceedings of the 26th International Joint Conference on Artificial Intelligence. 2017, 729–735

  7. Bessière C, Régin J C. MAC and combined heuristics: two reasons to forsake FC (and CBJ?) on hard problems. In: Proceedings of the 2nd International Conference on Principles and Practice of Constraint Programming. 1996, 61–75

  8. Chen X, Van B P. Conflict-directed backjumping revisited. Journal of Artificial Intelligence Research, 2001, 14(1): 53–81

    Article  MathSciNet  Google Scholar 

  9. Geelen P A. Dual viewpoint heuristics for binary constraint satisfaction problems. In: Proceedings of the 10th European Conference on Artificial Intelligence. 1992, 31–35

  10. Gent I P, MacIntyre E, Prosser P, Smith B M, Walsh T. An empirical study of dynamic variable ordering heuristics for the constraint satisfaction problem. In: Proceedings of the 2nd International Conference on Principles and Practice of Constraint Programming. 1996, 179–193

  11. Smith B M, Grant S A. Trying harder to fail first. In: Proceedings of the 13th European Conference on Artificial Intelligence. 1998, 41–55

  12. Nightingale P. Consistency for quantified constraint satisfaction problems. In: Proceedings of the 11th International Conference on Principles and Practice of Constraint Programming. 2005, 792–796

  13. Gent I P, Nightingale P, Rowley A, Stergiou K. Solving quantified constraint satisfaction problems. Artificial Intelligence, 2008, 17(6–7): 738–771

    Article  MathSciNet  Google Scholar 

  14. Gent I P, Nightingale P, Stergiou K. QCSP-Solve: a solver for quantified constraint satisfaction problems. In: Proceedings of the 19th International Joint Conference on Artificial Intelligence. 2005, 138–143

  15. Büning H K, Karpinski M, Flögel A. Resolution for quantified boolean formulas. Information and Computation, 1995, 117(1): 12–18

    Article  MathSciNet  Google Scholar 

  16. Nightingale P. Non-binary quantified CSP: algorithms and modelling. Constraints, 2009, 14(4): 539–581

    Article  MathSciNet  Google Scholar 

  17. Benedetti M, Lallouet A, Vautard J. Modeling adversary scheduling with QCSP+. In: Proceedings of the 2008 ACM Symposium on Applied Computing. 2008, 151–155

  18. Verger G, Bessière C. Guiding search in QCSP+ with back-propagation. In: Proceedings of the 14th International Conference on Principles and Practice of Constraint Programming. 2008, 175–189

  19. Verger G, Bessière C. A bottom-up approach for solving quantified CSPs. In: Proceedings of the 12th International Conference on Principles and Practice of Constraint Programming. 2006, 635–649

  20. Lallouet A, Lee J, Mak T W. Consistencies for ultra-weak solutions in minimax weighted CSPs using the duality principle. In: Proceedings of the 18th International Conference on Principles and Practice of Constraint Programming. 2012, 373–389

  21. Lallouet A, Lee J, Mak T W, Yip J. Ultra-weak solutions and consistency enforcement in minimax weighted constraint satisfaction. Constraints, 2015, 20(2): 109–154

    Article  MathSciNet  Google Scholar 

  22. Li H, Li Z. A novel strategy of combining variable ordering heuristics for constraint satisfaction problems. IEEE Access, 2018, 6: 42750–42756

    Article  Google Scholar 

  23. Li H, Liang Y, Zhang N, Guo J, Xu D, Li Z. Improving degree-based variable ordering heuristics for solving constraint satisfaction problems. Journal of Heuristics, 2016, 22(2): 125–145

    Article  Google Scholar 

  24. Stergiou K. Preprocessing quantified constraint satisfaction problems with value reordering and directional arc and path consistency. International Journal on Artificial Intelligence Tools, 2008, 17(2): 321–337

    Article  Google Scholar 

  25. Bessière C, Régin J C, Yap R H, Zhang Y. An optimal coarse-grained arc consistency algorithm. Artificial Intelligence, 2015, 165(2): 165–185

    Article  MathSciNet  Google Scholar 

  26. Prosser P. Hybrid algorithms for the constraint satisfaction problem. Computational Intelligence, 1993, 9(3): 268–299

    Article  Google Scholar 

  27. Davis M, Putnam H. A computing procedure for quantification theory. Journal of the ACM, 1960, 7(3): 201–215

    Article  MathSciNet  Google Scholar 

  28. Davis M, Logemann G, Loveland D. A machine program for theorem proving. Communications of the ACM, 1962, 5(7): 394–397

    Article  MathSciNet  Google Scholar 

  29. Bacchus F, Stergiou K. Solution directed backjumping for QCSP. In: Proceedings of the 13th International Conference on Principles and Practice of Constraint Programming. 2007, 148–163

  30. Stynes D, Brown K N. Value ordering for quantified CSPs. Constraints, 2009, 14(1): 16–37

    Article  MathSciNet  Google Scholar 

  31. Achlioptas D, Kirousis L M, Kranakis E, Krizanc D, Molloy M, Stamatiou Y C. Random constraint satisfaction: a more accurate picture. In: Proceedings of the 3rd International Conference on Principles and Practice of Constraint Programming. 1997, 107–120

  32. Xu K, Li W. Exact phase transitions in random constraint satisfaction problems. Journal of Artificial Intelligence Research, 2000, 12(1): 93–103

    Article  MathSciNet  Google Scholar 

  33. Xu K, Li W. An average analysis of backtracking on random constraint satisfaction problems. Annals of Mathematics and Artificial Intelligence, 2001, 33(1): 21–37

    Article  MathSciNet  Google Scholar 

  34. Xu K, Li W. Many hard examples in exact phase transitions. Theoretical Computer Science, 2006, 355(1): 291–302

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

We would like to thank Dr. Peter Nightingale for the source code of QCSP-Solve. The work described in this paper was supported by the National Natural Science Foundation of China (Granted Nos. 61972063, 61763003, 61672122, 61602077, 61402070), and the Fundamental Research Funds for the Central Universities (3132019029, 3132019355).

Author information

Authors and Affiliations

  1. College of Information Science and Technology, Dalian Maritime University, Dalian, 116026, China

    Jian Gao, Kuixian Wu & Rong Chen

  2. Logistics Research Institute, Dalian Maritime University, Dalian, 116026, China

    Jian Gao

  3. College of Computer Science and Information Engineering, Guangxi Normal University, Guilin, 541004, China

    Jinyan Wang

Authors
  1. Jian Gao
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jinyan Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Kuixian Wu
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Rong Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Rong Chen.

Additional information

Jian Gao received the PhD degree in Computer Application Technology from Dalian Maritime University, China. He is currently an associate professor of software engineering in Dalian Maritime University, China. His research interests include constraint programming and automatic reasoning.

Jinyan Wang received the MS and PhD degrees in Computer Application Technology and Operational Research and Cybernetics from Northeast Normal University, China, respectively. She is currently an associate professor of College of Computer Science and Information Technology in Guangxi Normal University, China. Her research interests include data security, uncertainty theory, and automatic reasoning.

Kuixian Wu is currently pursuing the MS degree in Computer Science and Technology from Dalian Maritime University, China. His research interest is combinatorial optimization.

Rong Chen received the MS and PhD degrees in Computer Software and Theory from Jilin University, China in 1997 and 2000. He is currently a professor of Dalian Maritime University, and has previously held position at Sun Yat-sen University, China. His research interests are in software diagnosis, collective intelligence, activity recognition, Internet and mobile computing.

Electronic supplementary material

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Gao, J., Wang, J., Wu, K. et al. Solving quantified constraint satisfaction problems with value selection rules. Front. Comput. Sci. 14, 145317 (2020). https://doi.org/10.1007/s11704-019-9179-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s11704-019-9179-9

Keywords

Search

Navigation