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Parameterized Complexity and Kernel Bounds for Hard Planning Problems

  • Christer Bäckström
  • Peter Jonsson
  • Sebastian Ordyniak
  • Stefan Szeider
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7878)

Abstract

The propositional planning problem is a notoriously difficult computational problem. Downey et al. (1999) initiated the parameterized analysis of planning (with plan length as the parameter) and Bäckström et al. (2012) picked up this line of research and provided an extensive parameterized analysis under various restrictions, leaving open only one stubborn case. We continue this work and provide a full classification. In particular, we show that the case when actions have no preconditions and at most e postconditions is fixed-parameter tractable if e ≤ 2 and W[1]-complete otherwise. We show fixed-parameter tractability by a reduction to a variant of the Steiner Tree problem; this problem has been shown fixed-parameter tractable by Guo et al. (2007). If a problem is fixed-parameter tractable, then it admits a polynomial-time self-reduction to instances whose input size is bounded by a function of the parameter, called the kernel. For some problems, this function is even polynomial which has desirable computational implications. Recent research in parameterized complexity has focused on classifying fixed-parameter tractable problems on whether they admit polynomial kernels or not. We revisit all the previously obtained restrictions of planning that are fixed-parameter tractable and show that none of them admits a polynomial kernel unless the polynomial hierarchy collapses to its third level.

Keywords

Goal State Parameterized Complexity Parameterized Problem Polynomial Kernel Kernel Size 
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.
    Bäckström, C., Chen, Y., Jonsson, P., Ordyniak, S., Szeider, S.: The complexity of planning revisited - a parameterized analysis. In: Hoffmann, J., Selman, B. (eds.) Proceedings of the Twenty-Sixth AAAI Conference on Artificial Intelligence, Toronto, Ontario, Canada, July 22-26. AAAI Press (2012)Google Scholar
  2. 2.
    Bäckström, C., Klein, I.: Planning in polynomial time: the SAS-PUBS class. Comput. Intelligence 7, 181–197 (1991)CrossRefGoogle Scholar
  3. 3.
    Bäckström, C., Nebel, B.: Complexity results for SAS+ planning. Comput. Intelligence 11, 625–656 (1995)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Bylander, T.: The computational complexity of propositional STRIPS planning. Artificial Intelligence 69(1-2), 165–204 (1994)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Bodlaender, H.L., Downey, R.G., Fellows, M.R., Hermelin, D.: On problems without polynomial kernels. J. of Computer and System Sciences 75(8), 423–434 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Bodlaender, H.L., Thomassé, S., Yeo, A.: Kernel bounds for disjoint cycles and disjoint paths. In: Fiat, A., Sanders, P. (eds.) ESA 2009. LNCS, vol. 5757, pp. 635–646. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  7. 7.
    Chen, J., Huang, X., Kanj, I.A., Xia, G.: Strong computational lower bounds via parameterized complexity. J. of Computer and System Sciences 72(8), 1346–1367 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Downey, R., Fellows, M.R., Stege, U.: Parameterized complexity: A framework for systematically confronting computational intractability. In: Contemporary Trends in Discrete Mathematics: From DIMACS and DIMATIA to the Future. AMS-DIMACS, vol. 49, pp. 49–99. American Mathematical Society (1999)Google Scholar
  9. 9.
    Downey, R.G., Fellows, M.R.: Parameterized Complexity. Monographs in Computer Science. Springer, New York (1999)Google Scholar
  10. 10.
    Fellows, M.R.: The lost continent of polynomial time: Preprocessing and kernelization. In: Bodlaender, H.L., Langston, M.A. (eds.) IWPEC 2006. LNCS, vol. 4169, pp. 276–277. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  11. 11.
    Flum, J., Grohe, M.: Parameterized Complexity Theory. Texts in Theoretical Computer Science. An EATCS Series, vol. XIV. Springer (2006)Google Scholar
  12. 12.
    Fomin, F.V.: Kernelization. In: Ablayev, F., Mayr, E.W. (eds.) CSR 2010. LNCS, vol. 6072, pp. 107–108. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  13. 13.
    Ghallab, M., Nau, D.S., Traverso, P.: Automated planning - theory and practice. Elsevier (2004)Google Scholar
  14. 14.
    Guo, J., Niedermeier, R.: Invitation to data reduction and problem kernelization. ACM SIGACT News 38(2), 31–45 (2007)CrossRefGoogle Scholar
  15. 15.
    Guo, J., Niedermeier, R., Suchý, O.: Parameterized complexity of arc-weighted directed steiner problems. SIAM J. Discrete Math. 25(2), 583–599 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Pietrzak, K.: On the parameterized complexity of the fixed alphabet shortest common supersequence and longest common subsequence problems. J. of Computer and System Sciences 67(4), 757–771 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Yap, C.-K.: Some consequences of nonuniform conditions on uniform classes. Theoretical Computer Science 26(3), 287–300 (1983)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Christer Bäckström
    • 1
  • Peter Jonsson
    • 1
  • Sebastian Ordyniak
    • 2
  • Stefan Szeider
    • 3
  1. 1.Department of Computer ScienceLinköping UniversityLinköpingSweden
  2. 2.Faculty of InformaticsMasaryk UniversityBrnoCzech Republic
  3. 3.Institute of Information SystemsVienna University of TechnologyViennaAustria

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