Abstract
By Courcelle’s Theorem we know that any property of finite structures definable in monadic second-order logic (MSO) becomes tractable over structures with bounded treewidth. This result was extended to counting problems by Arnborg et al. and to enumeration problems by Flum et al. Despite the undisputed importance of these results for proving fixed-parameter tractability, they do not directly yield implementable algorithms. Recently, Gottlob et al. presented a new approach using monadic datalog to close the gap between theoretical tractability and practical computability for MSO-definable decision problems. In the current work we show how counting and enumeration problems can be tackled by an appropriate extension of the datalog approach.
Supported by the Austrian Science Fund (FWF), project P20704-N18.
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References
Afrati, F.N., Chirkova, R.: Selecting and using views to compute aggregate queries. In: Eiter, T., Libkin, L. (eds.) ICDT 2005. LNCS, vol. 3363, pp. 383–397. Springer, Heidelberg (2004)
Arnborg, S., Lagergren, J., Seese, D.: Easy problems for tree-decomposable graphs. Journal of Algorithms 12(2), 308–340 (1991)
Bagan, G.: MSO queries on tree decomposable structures are computable with linear delay. In: Ésik, Z. (ed.) CSL 2006. LNCS, vol. 4207, pp. 167–181. Springer, Heidelberg (2006)
Bodlaender, H.L.: A linear-time algorithm for finding tree-decompositions of small treewidth. SIAM J. Comput. 25(6), 1305–1317 (1996)
Ceri, S., Gottlob, G., Tanca, L.: Logic Programming and Databases. Springer, Heidelberg (1990)
Cohen, S., Nutt, W., Serebrenik, A.: Rewriting aggregate queries using views. In: Proc. PODS 1999, pp. 155–166. ACM, New York (1999)
Courcelle, B.: Graph rewriting: An algebraic and logic approach. In: Handbook of Theoretical Computer Science, vol. B, pp. 193–242. Elsevier Science Publishers, Amsterdam (1990)
Courcelle, B.: Linear delay enumeration and monadic second-order logic. Discrete Applied Mathematics 157(12), 2675–2700 (2009)
Downey, R.G., Fellows, M.R.: Parameterized Complexity. Springer, New York (1999)
Eiter, T., Faber, W., Fink, M., Woltran, S.: Complexity results for answer set programming with bounded predicate arities and implications. Annals of Mathematics and Artificial Intelligence 51(2-4), 123–165 (2007)
Flum, J., Frick, M., Grohe, M.: Query evaluation via tree-decompositions. Journal of the ACM 49(6), 716–752 (2002)
Flum, J., Grohe, M.: Parameterized Complexity Theory. Springer, Heidelberg (2006)
Frick, M., Grohe, M.: The complexity of first-order and monadic second-order logic revisited. In: Proc. LICS 2002, pp. 215–224 (2002)
Gottlob, G., Pichler, R., Wei, F.: Monadic datalog over finite structures with bounded treewidth. In: Proc. PODS 2007, pp. 165–174. ACM, New York (2007)
Grohe, M.: Descriptive and parameterized complexity. In: Flum, J., Rodríguez-Artalejo, M. (eds.) CSL 1999. LNCS, vol. 1683, pp. 14–31. Springer, Heidelberg (1999)
Grumbach, S., Rafanelli, M., Tininini, L.: On the equivalence and rewriting of aggregate queries. Acta Inf. 40(8), 529–584 (2004)
Jakl, M., Pichler, R., Rümmele, S., Woltran, S.: Fast counting with bounded treewidth. In: Cervesato, I., Veith, H., Voronkov, A. (eds.) LPAR 2008. LNCS (LNAI), vol. 5330, pp. 436–450. Springer, Heidelberg (2008)
Kemp, D.B., Stuckey, P.J.: Semantics of logic programs with aggregates. In: Proc. ISLP, pp. 387–401 (1991)
Kloks, T.: Treewidth. LNCS, vol. 842. Springer, Heidelberg (1994)
Klug, A.C.: Equivalence of relational algebra and relational calculus query languages having aggregate functions. J. ACM 29(3), 699–717 (1982)
Libkin, L.: Elements of Finite Model Theory. Springer, Heidelberg (2004)
Szeider, S.: Monadic second order logic on graphs with local cardinality constraints. In: Ochmański, E., Tyszkiewicz, J. (eds.) MFCS 2008. LNCS, vol. 5162, pp. 601–612. Springer, Heidelberg (2008)
Vardi, M.Y.: The complexity of relational query languages (extended abstract). In: Proc. STOC 1982, pp. 137–146. ACM, New York (1982)
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Pichler, R., Rümmele, S., Woltran, S. (2010). Counting and Enumeration Problems with Bounded Treewidth. In: Clarke, E.M., Voronkov, A. (eds) Logic for Programming, Artificial Intelligence, and Reasoning. LPAR 2010. Lecture Notes in Computer Science(), vol 6355. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-17511-4_22
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DOI: https://doi.org/10.1007/978-3-642-17511-4_22
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