Advertisement

SAT-Based Local Improvement for Finding Tree Decompositions of Small Width

  • Johannes K. Fichte
  • Neha Lodha
  • Stefan Szeider
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10491)

Abstract

Many hard problems can be solved efficiently for problem instances that can be decomposed by tree decompositions of small width. In particular for problems beyond NP, such as #P-complete counting problems, tree decomposition-based methods are particularly attractive. However, finding an optimal tree decomposition is itself an NP-hard problem. Existing methods for finding tree decompositions of small width either (a) yield optimal tree decompositions but are applicable only to small instances or (b) are based on greedy heuristics which often yield tree decompositions that are far from optimal. In this paper, we propose a new method that combines (a) and (b), where a heuristically obtained tree decomposition is improved locally by means of a SAT encoding. We provide an experimental evaluation of our new method.

References

  1. 1.
    Abseher, M., Musliu, N., Woltran, S.: htd – a free, open-source framework for (customized) tree decompositions and beyond. In: Salvagnin, D., Lombardi, M. (eds.) CPAIOR 2017. LNCS, vol. 10335, pp. 376–386. Springer, Cham (2017). doi: 10.1007/978-3-319-59776-8_30 CrossRefGoogle Scholar
  2. 2.
    Arnborg, S., Corneil, D.G., Proskurowski, A.: Complexity of finding embeddings in a \(k\)-tree. SIAM J. Algebraic Discrete Methods 8(2), 277–284 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bannach, M., Berndt, S., Ehlers, T.: Jdrasil: a modular library for computing tree decompositions. Technical report, Lübeck University, Germany (2016)Google Scholar
  4. 4.
    Berg, J., Järvisalo, M.: SAT-based approaches to treewidth computation: an evaluation. In: Proceedings of the 26th IEEE International Conference on Tools with Artificial Intelligence, ICTAI 2014, pp. 328–335. IEEE Computer Society, Limassol, Cyprus, November 2014Google Scholar
  5. 5.
    Bodlaender, H.L., Koster, A.M.C.A.: Combinatorial optimization on graphs of bounded treewidth. Comput. J. 51(3), 255–269 (2008)CrossRefGoogle Scholar
  6. 6.
    Bodlaender, H.L., Koster, A.M.C.A.: Treewidth computations I. Upper bounds. Inf. Comput. 208(3), 259–275 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    van den Broek, J.W., Bodlaender, H.: TreewidthLIB - a benchmark for algorithms for treewidth and related graph problems. Technical report, Faculty of Science, Utrecht University (2010). http://www.staff.science.uu.nl/~bodla101/treewidthlib/
  8. 8.
    Chimani, M., Mutzel, P., Zey, B.: Improved Steiner tree algorithms for bounded treewidth. J. Discrete Algorithms 16, 67–78 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Courcelle, B., Makowsky, J.A., Rotics, U.: On the fixed parameter complexity of graph enumeration problems definable in monadic second-order logic. Discr. Appl. Math. 108(1–2), 23–52 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Darwiche, A.: A differential approach to inference in Bayesian networks. J. ACM 50(3), 280–305 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Dechter, R.: Tractable structures for constraint satisfaction problems. In: Rossi, F., van Beek, P., Walsh, T. (eds.) Handbook of Constraint Programming, Chap. 7, vol. I, pp. 209–244. Elsevier, Amsterdam (2006)Google Scholar
  12. 12.
    Dechter, R.: Graphical model algorithms at UC Irvine. Technical report, UC Irvine (2013). The network instances consist of Bayesian and Markov network susedin UAI competition and protein folding/side-chain prediction problems. http://graphmod.ics.uci.edu/group
  13. 13.
    Dell, H., Rosamond, F.: The parameterized algorithms and computational experiments challenge (2016). https://pacechallenge.wordpress.com/
  14. 14.
    Fichte, J.K.: daajoe/gtfs2graphs - a GTFS transit feed to graph format converter (2016). https://github.com/daajoe/gtfs2graphs
  15. 15.
    Fichte, J.K., Lodha, N., Szeider, S.: Trellis: treewidth local improvement solver (2017). https://github.com/daajoe/trellis
  16. 16.
    Freuder, E.C.: A sufficient condition for backtrack-bounded search. J. ACM 32(4), 755–761 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Gaspers, S., Gudmundsson, J., Jones, M., Mestre, J., Rümmele, S.: Turbocharging Treewidth Heuristics. In: Guo, J., Hermelin, D. (eds.) 11th International Symposium on Parameterized and Exact Computation (IPEC 2016). Leibniz International Proceedings in Informatics (LIPIcs), vol. 63, pp. 13:1–13:13. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik, Dagstuhl, Germany (2017)Google Scholar
  18. 18.
    Gogate, V., Dechter, R.: A complete anytime algorithm for treewidth. In: Proceedings of the Twentieth Conference Annual Conference on Uncertainty in Artificial Intelligence (UAI 2004), pp. 201–208. AUAI Press, Arlington (2004)Google Scholar
  19. 19.
    Gottlob, G., Pichler, R., Wei, F.: Bounded treewidth as a key to tractability of knowledge representation and reasoning. Artif. Intell. 174(1), 105–132 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Hammerl, T., Musliu, N., Schafhauser, W.: Metaheuristic algorithms and tree decomposition. In: Kacprzyk, J., Pedrycz, W. (eds.) Springer Handbook of Computational Intelligence, pp. 1255–1270. Springer, Heidelberg (2015). doi: 10.1007/978-3-662-43505-2_64 CrossRefGoogle Scholar
  21. 21.
    Kask, K., Gelfand, A., Otten, L., Dechter, R.: Pushing the power of stochastic greedy ordering schemes for inference in graphical models. In: Burgard, W., Roth, D. (eds.) Proceedings of the Twenty-Fifth AAAI Conference on Artificial Intelligence, AAAI 2011. AAAI Press (2011)Google Scholar
  22. 22.
  23. 23.
    Kloks, T.: Treewidth: Computations and Approximations. Springer, Heidelberg (1994)CrossRefzbMATHGoogle Scholar
  24. 24.
    Lauritzen, S.L., Spiegelhalter, D.J.: Local computations with probabilities on graphical structures and their application to expert systems. J. Roy. Statist. Soc. Ser. B 50(2), 157–224 (1988)MathSciNetzbMATHGoogle Scholar
  25. 25.
    Lodha, N., Ordyniak, S., Szeider, S.: A SAT approach to branchwidth. In: Creignou, N., Le Berre, D. (eds.) SAT 2016. LNCS, vol. 9710, pp. 179–195. Springer, Cham (2016). doi: 10.1007/978-3-319-40970-2_12 Google Scholar
  26. 26.
    Ordyniak, S., Szeider, S.: Parameterized complexity results for exact Bayesian network structure learning. J. Artif. Intell. Res. 46, 263–302 (2013)MathSciNetzbMATHGoogle Scholar
  27. 27.
    Roussel, O.: Controlling a solver execution with the runsolver tool. J. Satisfiability Boolean Model. Comput. 7, 139–144 (2011)MathSciNetzbMATHGoogle Scholar
  28. 28.
    Samer, M., Veith, H.: Encoding treewidth into SAT. In: Kullmann, O. (ed.) SAT 2009. LNCS, vol. 5584, pp. 45–50. Springer, Heidelberg (2009). doi: 10.1007/978-3-642-02777-2_6 CrossRefGoogle Scholar
  29. 29.
    Sinz, C.: Towards an optimal CNF encoding of boolean cardinality constraints. In: Beek, P. (ed.) CP 2005. LNCS, vol. 3709, pp. 827–831. Springer, Heidelberg (2005). doi: 10.1007/11564751_73 CrossRefGoogle Scholar
  30. 30.
    Song, Y., Liu, C., Malmberg, R.L., Pan, F., Cai, L.: Tree decomposition based fast search of RNA structures including pseudoknots in genomes. In: Proceedings of the 4th International IEEE Computer Society Computational Systems Bioinformatics Conference, CSB 2005, pp. 223–234. IEEE Computer Society (2005)Google Scholar
  31. 31.
    Tamaki, H.: TCS-Meiji (2016). https://github.com/TCS-Meiji/treewidth-exact

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Johannes K. Fichte
    • 1
  • Neha Lodha
    • 1
  • Stefan Szeider
    • 1
  1. 1.TU WienViennaAustria

Personalised recommendations