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.
Research was supported by the Austrian Science Fund (FWF), Grants Y698, W1255-N23, and P-26696. The first author is also affiliated with the University of Potsdam, Germany.
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Notes
- 1.
Retrieved on March 26, 2017.
- 2.
The run and analysis tool is available online at https://github.com/daajoe/benchmark-tool. The file benchmark-tool/runscripts/treewidth/localimprovement. xml contains all solver flags to reproduce our benchmark runs.
References
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
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)
Bannach, M., Berndt, S., Ehlers, T.: Jdrasil: a modular library for computing tree decompositions. Technical report, Lübeck University, Germany (2016)
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 2014
Bodlaender, H.L., Koster, A.M.C.A.: Combinatorial optimization on graphs of bounded treewidth. Comput. J. 51(3), 255–269 (2008)
Bodlaender, H.L., Koster, A.M.C.A.: Treewidth computations I. Upper bounds. Inf. Comput. 208(3), 259–275 (2010)
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/
Chimani, M., Mutzel, P., Zey, B.: Improved Steiner tree algorithms for bounded treewidth. J. Discrete Algorithms 16, 67–78 (2012)
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)
Darwiche, A.: A differential approach to inference in Bayesian networks. J. ACM 50(3), 280–305 (2003)
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)
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
Dell, H., Rosamond, F.: The parameterized algorithms and computational experiments challenge (2016). https://pacechallenge.wordpress.com/
Fichte, J.K.: daajoe/gtfs2graphs - a GTFS transit feed to graph format converter (2016). https://github.com/daajoe/gtfs2graphs
Fichte, J.K., Lodha, N., Szeider, S.: Trellis: treewidth local improvement solver (2017). https://github.com/daajoe/trellis
Freuder, E.C.: A sufficient condition for backtrack-bounded search. J. ACM 32(4), 755–761 (1985)
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)
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)
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)
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
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)
Kittan, K.: Zuse cluster (2017). http://www.cs.uni-potsdam.de/bs/research/labsZuse.html
Kloks, T.: Treewidth: Computations and Approximations. Springer, Heidelberg (1994)
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)
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
Ordyniak, S., Szeider, S.: Parameterized complexity results for exact Bayesian network structure learning. J. Artif. Intell. Res. 46, 263–302 (2013)
Roussel, O.: Controlling a solver execution with the runsolver tool. J. Satisfiability Boolean Model. Comput. 7, 139–144 (2011)
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
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
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)
Tamaki, H.: TCS-Meiji (2016). https://github.com/TCS-Meiji/treewidth-exact
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Fichte, J.K., Lodha, N., Szeider, S. (2017). SAT-Based Local Improvement for Finding Tree Decompositions of Small Width. In: Gaspers, S., Walsh, T. (eds) Theory and Applications of Satisfiability Testing – SAT 2017. SAT 2017. Lecture Notes in Computer Science(), vol 10491. Springer, Cham. https://doi.org/10.1007/978-3-319-66263-3_25
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