Annals of Operations Research

, Volume 184, Issue 1, pp 27–50 | Cite as

New filtering for the cumulative constraint in the context of non-overlapping rectangles

  • Nicolas Beldiceanu
  • Mats Carlsson
  • Sophie Demassey
  • Emmanuel Poder
Article

Abstract

This article describes new filtering methods for the cumulative constraint. The first method introduces the so called longest closed hole and longest open hole problems. For these two problems it first provides bounds and exact methods and then shows how to use them in the context of the non-overlapping constraint. The second method introduces balancing knapsack constraints which relate the total height of the tasks that end at a specific time-point with the total height of the tasks that start at the same time-point. Experiments on tight rectangle packing problems show that these methods drastically reduce both the time and the number of backtracks for finding all solutions as well as for finding the first solution. For example, we found without backtracking all solutions to 65 perfect square instances of order 22–25 and sizes ranging from 192×192 to 661×661.

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References

  1. Aggoun, A., & Beldiceanu, N. (1993). Extending CHIP in order to solve complex scheduling and placement problems. Mathematical and Computer Modelling, 17(7), 57–73. CrossRefGoogle Scholar
  2. Baptiste, P., & Le Pape, C. (1999). Constraint propagation techniques for cumulative scheduling. In Proceedings of the first workshop on the integration of AI and OR techniques in constraint programming for combinatorial optimization problems, Ferrara, Italy, 1999. Google Scholar
  3. Beldiceanu, N., & Contejean, E. (1994). Introducing global constraints in CHIP. Mathematical and Computer Modelling, 20(12), 97–123. CrossRefGoogle Scholar
  4. Beldiceanu, N., Carlsson, M., Poder, E., Sadek, R., & Truchet, C. (2007). A generic geometrical constraint kernel in space and time for handling polymorphic k-dimensional objects. In C. Bessière (Ed.), LNCS : Vol. 4741. Proc. CP’2007 (pp. 180–194). Berlin: Springer. Google Scholar
  5. Biró, M. (1990). Object-oriented interaction in resource constrained scheduling. Information Processing Letters, 36(2), 65–67. CrossRefGoogle Scholar
  6. Bouwkamp, C. J., & Duijvestijn, A. J. W. (1992). Catalogue of simple perfect squared squares of orders 21 through 25. Technical report EUT Report 92-WSK-03, Eindhoven University of Technology, The Netherlands, November 1992. Google Scholar
  7. Caseau, Y., & Laburthe, F. (1996). Cumulative scheduling with task intervals. In Joint international conference and symposium on logic programming (JICSLP’96). Cambridge: MIT Press. Google Scholar
  8. Clautiaux, F., Carlier, J., & Moukrim, A. (2007). A new exact method for the two-dimensional orthogonal packing problem. European Journal of Operational Research, 183(3), 1196–1211. CrossRefGoogle Scholar
  9. Clautiaux, F., Jouglet, A., Carlier, J., & Moukrim, A. (2008). A new constraint programming approach for the orthogonal packing problem. Computers and Operation Research, 35(3), 944–959. CrossRefGoogle Scholar
  10. Erschler, J., & Lopez, P. (1990). Energy-based approach for task scheduling under time and resources constraints. In 2nd international workshop on project management and scheduling (pp. 115–121). Compiégne, France, June 1990. Google Scholar
  11. Garey, M. R., & Johnson, D. S. (1979). Computers and intractibility. A guide to the theory of NP-completeness. New York: Freeman. Google Scholar
  12. Lahrichi, A. (1982). Scheduling: the notions of hump, compulsory parts and their use in cumulative problems. Comptes Rendus de l’Académie Des Sciences, Paris, 294, 209–211. Google Scholar
  13. Lesh, N., Marks, J., McMahon, A., & Mitzenmacher, M. (2004). Exhaustive approaches to 2D rectangular perfect packings. Information Processing Letters, 90(1), 7–14. CrossRefGoogle Scholar
  14. Mercier, L., & Van Hentenryck, P. (2008). Edge-finding for cumulative scheduling. INFORMS Journal on Computing, 20(1), 143–153. CrossRefGoogle Scholar
  15. Simonis, H., & O’Sullivan, B. (2008). Search strategies for rectangle packing. In P. J. Stuckey (Ed.), LNCS : Vol. 5202. Proc. CP’2008 (pp. 52–66). Berlin: Springer. Google Scholar
  16. Trick, M. A. (2003). A dynamic programming approach for consistency and propagation for knapsack constraints. Annals of Operations Research, 118(1–4), 73–84. CrossRefGoogle Scholar
  17. Van Hentenryck, P. (1994). Scheduling and packing in the constraint language cc(FD). In M. Zweben & M. Fox (Eds.), Intelligent scheduling. San Mateo: Morgan Kaufmann. Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  • Nicolas Beldiceanu
    • 1
  • Mats Carlsson
    • 2
  • Sophie Demassey
    • 1
  • Emmanuel Poder
    • 1
  1. 1.École des Mines de NantesLINA UMR CNRS 6241NantesFrance
  2. 2.SICSKistaSweden

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