Hostname: page-component-8448b6f56d-c47g7 Total loading time: 0 Render date: 2024-04-15T12:56:24.962Z Has data issue: false hasContentIssue false
  • English
  • Français

Tractable answer-set programming with weight constraints: bounded treewidth is not enough*

Published online by Cambridge University Press:  17 July 2012

REINHARD PICHLER ,
STEFAN RÜMMELE ,
STEFAN SZEIDER  and
STEFAN WOLTRAN
REINHARD PICHLER
Affiliation:
Vienna University of Technology, Austria (e-mail: pichler@dbai.tuwien.ac.at, ruemmele@dbai.tuwien.ac.at, woltran@dbai.tuwien.ac.at, stefan@szeider.net)
STEFAN RÜMMELE
Affiliation:
Vienna University of Technology, Austria (e-mail: pichler@dbai.tuwien.ac.at, ruemmele@dbai.tuwien.ac.at, woltran@dbai.tuwien.ac.at, stefan@szeider.net)
STEFAN SZEIDER
Affiliation:
Vienna University of Technology, Austria (e-mail: pichler@dbai.tuwien.ac.at, ruemmele@dbai.tuwien.ac.at, woltran@dbai.tuwien.ac.at, stefan@szeider.net)
STEFAN WOLTRAN
Affiliation:
Vienna University of Technology, Austria (e-mail: pichler@dbai.tuwien.ac.at, ruemmele@dbai.tuwien.ac.at, woltran@dbai.tuwien.ac.at, stefan@szeider.net)
Get access Rights & Permissions [Opens in a new window]

Abstract

Cardinality constraints or, more generally, weight constraints are well recognized as an important extension of answer-set programming. Clearly, all common algorithmic tasks related to programs with cardinality or weight constraints – like checking the consistency of a program – are intractable. Many intractable problems in the area of knowledge representation and reasoning have been shown to become linear time tractable if the treewidth of the programs or formulas under consideration is bounded by some constant. The goal of this paper is to apply the notion of treewidth to programs with cardinality or weight constraints and to identify tractable fragments. It will turn out that the straightforward application of treewidth to such class of programs does not suffice to obtain tractability. However, by imposing further restrictions, tractability can be achieved.

Keywords

answer-set programmingparameterized complexity analysistreewidth
Type
Regular Papers
Information
Theory and Practice of Logic Programming , Volume 14 , Issue 2 , March 2014 , pp. 141 - 164
Copyright
Copyright © Cambridge University Press 2012 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

*

A preliminary version appeared in the Proceedings of the Twelfth International Conference on Principles of Knowledge Representation and Reasoning (KR 2010).

References

Bodlaender, H. L. 1996. A linear-time algorithm for finding tree-decompositions of small treewidth. SIAM Journal on Computing 25, 6, 1305–1317.CrossRefGoogle Scholar
Bodlaender, H. L. and Koster, A. M. C. A. 2006. Safe separators for treewidth. Discrete Mathematics 306, 3, 337–350.CrossRefGoogle Scholar
Bodlaender, H. L. and Koster, A. M. C. A. 2008. Combinatorial optimization on graphs of bounded treewidth. Computer Journal 51, 3, 255–269.CrossRefGoogle Scholar
Courcelle, B. 1987. Recognizability and Second-Order Definability for Sets of Finite Graphs. Technical rep. I-8634, Université de Bordeaux, Bordeaux, France.Google Scholar
Dermaku, A., Ganzow, T., Gottlob, G., McMahan, B. J., Musliu, N. and Samer, M. 2008. Heuristic methods for hypertree decomposition. In Proceedings of the MICAI, New York, September 6–10. LNCS, vol. 5317. Springer, New York, USA, 1–11.Google Scholar
Downey, R. G. and Fellows, M. R. 1999. Parameterized Complexity. Springer-Verlag, Berlin, Germany.CrossRefGoogle Scholar
Flum, J. and Grohe, M. 2006. Parameterized Complexity Theory. Springer-Verlag, Berlin, Germany.Google Scholar
Gottlob, G., Pichler, R. and Wei, F. 2010. Bounded treewidth as a key to tractability of knowledge representation and reasoning. Artificial Intelligence 174, 1, 105–132.CrossRefGoogle Scholar
Jakl, M., Pichler, R. and Woltran, S. 2009. Answer-set programming with bounded treewidth. In Proceedings of the IJCAI '09, Pasadena, CA, USA, Boutilier, C., Ed., AAAI Press, Palo Alto, CA, 816–822.Google Scholar
Kask, K., Gelfand, A., Otten, L. and Dechter, R. 2011. Pushing the power of stochastic greedy ordering schemes for inference in graphical models. In Proceedings of the AAAI '11, Burgard, W. and Roth, D., Eds. AAAI Press, Palo Alto, CA, 54–60.Google Scholar
Kloks, T. 1994. Treewidth, Computations and Approximations. Springer-Verlag, Berlin, Germany.Google Scholar
Koster, A. M. C. A., Bodlaender, H. L. and van Hoesel, S. P. M. 2001. Treewidth: Computational experiments. Electronic Notes in Discrete Mathematics 8, 54–57.CrossRefGoogle Scholar
Liu, G. 2009. Level mapping-induced loop formulas for weight constraint and aggregate programs. In Proceedings of the LPNMR '09, Erdem, E., Lin, F. and Schaub, T., Eds. LNCS, vol. 5753. Springer, New York, 444–449.Google Scholar
Marek, V. W. and Truszczyński, M. 1999. Stable models and an alternative logic programming paradigm. In The Logic Programming Paradigm: A 25-Year Perspective, Apt, K. R., Marek, V. W., Truszczyński, M. and Warren, D. S., Eds. Springer, New York, 375–398.CrossRefGoogle Scholar
Niedermeier, R. 2006. Invitation to Fixed-Parameter Algorithms. Oxford University Press, Oxford, UK.CrossRefGoogle Scholar
Niemelä, I., Simons, P. and Soininen, T. 1999. Stable model semantics of weight constraint rules. In Proceedings of the LPNMR'99, Gelfond, M., Leone, N. and Pfeifer, G., Eds., LNCS, vol. 1730. Springer, New York, 317–331.Google Scholar
Szeider, S. 2010. Not so easy problems for tree decomposable graphs. In Advances in Discrete Mathematics and Applications: Mysore, 2008, Acharya, B. D., Katona, G. O. H. and Nesetril, Juarez, Eds. Ramanujan Mathematical Society Lecture Notes Series, vol. 13., Ramanujan Mathematical Society, Mysore, India, 179–190.Google Scholar
Szeider, S. 2011. Monadic second-order logic on graphs with local cardinality constraints. ACM Transactions on Computer Logic 12, 2, art. 12, 21.Google Scholar
van den Eijkhof, F., Bodlaender, H. L. and Koster, A. M. C. A. 2007. Safe reduction rules for weighted treewidth. Algorithmica 47, 2, 139–158.CrossRefGoogle Scholar