Skip to main content
Log in

Constraint-directed search for all-interval series

  • Published:
Constraints Aims and scope Submit manuscript

Abstract

All-interval series is a standard benchmark problem for constraint satisfaction search. An all-interval series of size n is a permutation of integers [0, n) such that the differences between adjacent integers are a permutation of [1, n). Generating each such all-interval series of size n is an interesting challenge for constraint community. The problem is very difficult in terms of the size of the search space. Different approaches have been used to date to generate all the solutions of AIS but the search space that must be explored still remains huge. In this paper, we present a constraint-directed backtracking-based tree search algorithm that performs efficient lazy checking rather than immediate constraint propagation. Moreover, we prove several key properties of all-interval series that help prune the search space significantly. The reduced search space essentially results into fewer backtracking. We also present scalable parallel versions of our algorithm that can exploit the advantage of having multi-core processors and even multiple computer systems. Our new algorithm generates all the solutions of size up to 27 while a satisfiability-based state-of-the-art approach generates all solutions up to size 24.

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

Access this article

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

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5

Similar content being viewed by others

References

  1. Ilog solver. http://www.cs.cornell.edu/w8/iisi/ilog/cp11/usrsolver/usrsolverpreface.html.

  2. Minizinc challenge 2013 results. https://www.minizinc.org/challenge2013/results2013.html.

  3. Adamaszek, M. (2006). Efficient enumeration of graceful permutations. arXiv:math/0608513.

  4. Alsinet, T., Bejar, R., Cabiscol, A., Fernandez, C., & Manya, F. (2002). Minimal and redundant SAT encodings for the all-interval series problem. In Topics in artificial intelligence, 5th catalonian conference on AI (CCIA), LNCS, (Vol. 2504 pp. 139–144).

  5. Beldiceanu, N., & Contejean, E. (1994). Introducing global constraints in CHIP. Journal of Mathematical and Computer Modelling, 20, 97–123.

    Article  MATH  Google Scholar 

  6. Béjar, R., Manyà, F., Cabiscol, A., Fernàndez, C., & Gomes, C. (2007). Regular-sat: a many-valued approach to solving combinatorial problems. Discrete Applied Mathematics, 155(12), 1613–1626.

    Article  MathSciNet  MATH  Google Scholar 

  7. Choi, C., & Lee, J. (2002). On the pruning behaviour of minimal combined models for permutation csps. In International workshop on reformulating constraint satisfaction problems.

  8. Codognet, P., & Diaz, D. (2001). Yet another local search method for constraint solving. In Proceedings of the international symposium on stochastic algorithms: foundations and applications, SAGA ’01 (pp. 73–90): Springer.

  9. Colles, H. (1940). Grove’s dictionary of music and musicians. New York: The MacMillan Company.

    Google Scholar 

  10. Dent, M.J., & Mercer, R.E. (1994). Minimal forward checking. In Proceedings of the 6th international conference on tools with artificial intelligence. (pp. 432–438): IEEE.doi:10.1109/TAI.1994.346460

  11. Gebser, M., Kaufmann, B., & Schaub, T. (2009). The conflict-driven answer set solver Clasp. Lecture Notes in Computer Science, Springer, 5753, 509–514.

    Article  Google Scholar 

  12. Gent, I. P., McDonald, I., & Smith, B. M. (2003). Conditional symmetry in the all-interval series problem. In Proceedings of the 3rd international workshop on symmetry in constraint satisfaction problems, (Vol. 3 pp. 55–65).

  13. Hoos, H. H. (1998). Stochastic local search - methods, models, applications. Ph.D. thesis, Department of Computer Science, Technical University of Darmstadt, Germany.

  14. Hudson, S. (1991). Incremental attribute evaluation: a flexible algorithm for lazy update. ACM Transactions on Programming Language and Systems, 13(3), 315–341. doi:10.1145/117009.117012.

    Article  Google Scholar 

  15. Krenek, E. (1974). Horizons circled-reflections on my music. Berkeley: University of California Press.

    Google Scholar 

  16. Moisan, T., Gaudreault, J., & Quimper, C. G. (2013). Parallel discrepancy-based search. In International conference on principles and practice of constraint programming (pp. 30–46): Springer.

  17. Newton, M., Pham, D., Sattar, A., & Maher, M. (2011). Kangaroo: an efficient constraint-based local search system using lazy propagation. CP. LNCS, Springer, Heidelberg, 6876, 645–659.

    Google Scholar 

  18. Nguyen, V., & Son, M. (2014). Solving the all-interval series problem: SAT vs CP. In Proceedings of the 5th symposium on information and communication technology (pp. 65–74).

  19. Petrie, K. E., & Smith, B. M. (2003). Symmetry breaking in graceful graphs. In Rossi, F. (Ed.) Proceedings of principles and practice of constraint programming: Springer.

  20. Puget, J., & Regin, J. Solving the all-interval problem. http://www.cs.st-andrews.ac.uk/ianm/CSPLib/prob/prob007/puget.pdf.

  21. Remzi, H.A.D., & Andrea, C.A.D. (2014). Operating systems: three easy steps Arpaci-Dusseau books.

  22. Schuurmans, D., & Southey, F. (2000). Local search characteristics of incomplete SAT procedures. In Proceedings of the 17th national conference on artificial intelligence (AAAI-2000) (pp. 297–302). Austin/TX, USA: AAAI Press.

    Google Scholar 

  23. Simonis, H., Beldiceanu, N., & Sa, C. (1998). A note on CSPLIB prob007. Tech. rep., Normale.

  24. Solnon, C. (2000). Solving permutation constraint satisfaction problems with artificial ants. In Proceedings of ECAI’2000 (pp. 118–122): IOS Press.

  25. Toro, M., Rueda, C., Agon, C., & Assayag, G. (2015). Gelisp: a library to represent musical csps and search strategies. Computing Research Repository. arXiv:1510.02828.

  26. Truchet, C., Richoux, F., & Codognet, P. (2012). Prediction of parallel speed-ups for las vegas algorithms. arXiv:1212.4287.

  27. Walsh, T. (2010). Parameterized complexity results in symmetry breaking. In Proceedings of 5th international symposium, IPEC, Chennai, India, LNCS, (Vol. 6478 pp. 4–13): Springer.

  28. Worister, M., Steinlechner, H., Maierhofer, S., & Tobler, R. (2013). Lazy incremental computation for efficient scene graph rendering. In Proceedings of the 5th high-performance graphics conference. doi:10.1145/2492045.2492051 (pp. 53–62). New York, NY, USA: ACM.

  29. Yadav, S.C., & Singh, S.K. (2009). An introduction to Client/Server Computing, New Age International Publishers Limited.

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Md Masbaul Alam Polash.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Polash, M.M.A., Newton, M.A.H. & Sattar, A. Constraint-directed search for all-interval series. Constraints 22, 403–431 (2017). https://doi.org/10.1007/s10601-016-9261-y

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10601-016-9261-y

Keywords

Navigation