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Solutions to Real-World Instances of PSPACE-Complete Stacking

  • Felix G. König
  • Macro Lübbecke
  • Rolf Möhring
  • Guido Schäfer
  • Ines Spenke
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4698)

Abstract

We investigate a complex stacking problem that stems from storage planning of steel slabs in integrated steel production. Besides the practical importance of such stacking tasks, they are appealing from a theoretical point of view. We show that already a simple version of our stacking problem is PSPACE-complete. Thus, fast algorithms for computing provably good solutions as they are required for practical purposes raise various algorithmic challenges. We describe an algorithm that computes solutions within 5/4 of optimality for all our real-world test instances. The basic idea is a search in an exponential state space that is guided by a state-valuation function. The algorithm is extremely fast and solves practical instances within a few seconds. We assess the quality of our solutions by computing instance-dependent lower bounds from a combinatorial relaxation formulated as mixed integer program. To the best of our knowledge, this is the first approach that provides quality guarantees for such problems.

Keywords

State Graph Valuation Function Permutation Graph Quantify Boolean Formula False Position 
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.
    Bertsekas, D.P., Tsitsiklis, J.N.: Neuro-Dynamic Programming. Athena Scientific (1996)Google Scholar
  2. 2.
    Bertsekas, D.P., Tsitsiklis, J.N., Wu, C.: Rollout algorithms for combinatorial optimization. Journal of Heuristics 3(3), 245–262 (1997)zbMATHCrossRefGoogle Scholar
  3. 3.
    Bonet, B., Loerincs, G., Geffner, H.: A robust and fast action selection mechanism for planning. In: Proceedings of the 14th National Conference on AI, pp. 714–719 (1997)Google Scholar
  4. 4.
    Cull, P., Ecklund, E.F.: Towers of hanoi and analysis of algorithms. The American Mathematical Monthly 90, 407–420 (1985)CrossRefGoogle Scholar
  5. 5.
    Dekker, R., Voogd, P., van Asperen, E.: Advanced methods for container stacking. OR Spectrum 28, 563–586 (2006)zbMATHCrossRefGoogle Scholar
  6. 6.
    Fox, M., Long, D.: Progress in AI planning research and application. CEPIS 3, 10–25 (2002)Google Scholar
  7. 7.
    Hansen, J., Clausen, J.: Crane scheduling for a plate storage. Technical Report 1, Informatics and Mathematical Modelling, Technical University of Denmark (2002)Google Scholar
  8. 8.
    Hearn, R.A., Demaine, E.D.: PSPACE-completeness of sliding-block puzzles and other problems through the nondeterministic constraint logic model of computation. Theoretical Computer Science 343(1–2), 72–96 (2005)zbMATHCrossRefGoogle Scholar
  9. 9.
    Jansen, K.: The mutual exclusion scheduling problem for permutation and comparability graphs. Information and Computation 180, 71–81 (2003)zbMATHCrossRefGoogle Scholar
  10. 10.
    McDermott, D.: A heuristic estimator for means ends analysis in planning. In: Proceedings of the 3rd International Conference on Artificial Intelligence Planning Systems, pp. 142–149 (1996)Google Scholar
  11. 11.
    Savitch, W.J.: Relationships between nondeterministic and deterministic tape complexities. Journal of Computer and System Sciences 4(2), 177–192 (1970)zbMATHGoogle Scholar
  12. 12.
    Steenken, D., Voß, S., Stahlbock, R.: Container terminal operation and operations research - a classification and literature review. OR Spectrum 26, 3–49 (2004)zbMATHCrossRefGoogle Scholar
  13. 13.
    Tang, L., Liu, J., Rong, A., Yang, Z.: Modelling and a genetic algorithm solution to the slab stack shuffling problem in implementing steel rolling schedulings. International Journal of Production Research 40(7), 1583–1595 (2002)zbMATHCrossRefGoogle Scholar
  14. 14.
  15. 15.
    Walsh, T.: The towers of hanoi revisited, moving the rings by counting the moves. Information Processing Letters 15(2), 64–67 (1982)zbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2007

Authors and Affiliations

  • Felix G. König
    • 1
  • Macro Lübbecke
    • 1
  • Rolf Möhring
    • 1
  • Guido Schäfer
    • 1
  • Ines Spenke
    • 1
  1. 1.Technische Universität Berlin, Institut für Mathematik, MA 6-1, Straße des 17. Juni 136, 10623 BerlinGermany

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