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Parameterized Compilation Lower Bounds for Restricted CNF-Formulas

  • Stefan Mengel
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9710)

Abstract

We show unconditional parameterized lower bounds in the area of knowledge compilation, more specifically on the size of circuits in decomposable negation normal form (DNNF) that encode CNF-formulas restricted by several graph width measures. In particular, we show that
  • there are CNF formulas of size n and modular incidence treewidth k whose smallest DNNF-encoding has size \(n^{\varOmega (k)}\), and

  • there are CNF formulas of size n and incidence neighborhood diversity k whose smallest DNNF-encoding has size \(n^{\varOmega (\sqrt{k})}\).

These results complement recent upper bounds for compiling CNF into DNNF and strengthen—quantitatively and qualitatively—known conditional lower bounds for cliquewidth. Moreover, they show that, unlike for many graph problems, the parameters considered here behave significantly differently from treewidth.

Notes

Acknowledgments

The author would like to thank Florent Capelli for helpful discussions. Moreover, he thanks Jean-Marie Lagniez for helpful discussions and for experiments with the compiler from [15].

References

  1. 1.
    Bova, S., Capelli, F., Mengel, S., Slivovsky, F.: On compiling CNFs into structured deterministic DNNFs. In: Heule, M., Weaver, S. (eds.) SAT 2015. LNCS, vol. 9340, pp. 199–214. Springer, Heidelberg (2015). doi: 10.1007/978-3-319-24318-4_15 CrossRefGoogle Scholar
  2. 2.
    Bova, S., Capelli, F., Mengel, S., Slivovsky, F.: Knowledge compilation meets communication complexity. In: Proceedings of the 25th International Joint Conference on Artificial Intelligence, IJCAI 2016. IJCAI/AAAI (to appear, 2016)Google Scholar
  3. 3.
    Brault-Baron, J., Capelli, F., Mengel, S.: Understanding model counting for beta-acyclic CNF-formulas. In: Mayr, E.W., Ollinger, N. (eds.) 32nd International Symposium on Theoretical Aspects of Computer Science, STACS 2015, Garching, Germany, 4–7 March 2015. LIPIcs, vol. 30, pp. 143–156. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik (2015)Google Scholar
  4. 4.
    Chen, H.: Parameterized compilability. In: Kaelbling, L.P., Saffiotti, A. (eds.) Proceedings of the Nineteenth International Joint Conference on Artificial Intelligence, IJCAI 2005, Edinburgh, Scotland, UK, 30 July–5 August 2005, pp. 412–417. Professional Book Center (2005)Google Scholar
  5. 5.
    Chen, H.: Parameter compilation. In: Husfeldt, T., Kanj, I.A. (eds.) 10th International Symposium on Parameterized and Exact Computation, IPEC 2015, Patras, Greece, 16–18 September 2015. LIPIcs, vol. 43, pp. 127–137. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik (2015)Google Scholar
  6. 6.
    Darwiche, A.: Decomposable negation normal form. J. ACM 48(4), 608–647 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Darwiche, A.: New advances in compiling CNF into decomposable negation normal form. In López de Mántaras, R., Saitta, L. (eds.) Proceedings of the 16th Eureopean Conference on Artificial Intelligence, ECAI 2004, including Prestigious Applicants of Intelligent Systems, PAIS 2004, Valencia, Spain, 22–27 August 2004, pp. 328–332. IOS Press (2004)Google Scholar
  8. 8.
    Darwiche, A., Marquis, P.: A knowledge compilation map. J. Artif. Intell. Res. (JAIR) 17, 229–264 (2002)MathSciNetzbMATHGoogle Scholar
  9. 9.
    de Haan, R.: An overview of non-uniform parameterized complexity. Electron. Colloquium Comput. Complex. (ECCC) 22, 130 (2015)Google Scholar
  10. 10.
    Dell, H., Kim, E.J., Lampis, M., Mitsou, V., Mömke, T.: Complexity and approximability of parameterized MAX-CSPs. In: Husfeldt, T., Kanj, I.A. (eds.) 10th International Symposium on Parameterized and Exact Computation, IPEC 2015, Patras, Greece, 16–18 September 2015. LIPIcs, vol. 43, pp. 294–306. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik (2015)Google Scholar
  11. 11.
    Diestel, R.: Graph Theory. Graduate Texts in Mathematics, vol. 173, 4th edn. Springer, Heidelberg (2012)zbMATHGoogle Scholar
  12. 12.
    Duris, P., Hromkovic, J., Jukna, S., Sauerhoff, M., Schnitger, G.: On multi-partition communication complexity. Inf. Comput. 194(1), 49–75 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Flum, J., Grohe, M.: Parameterized Complexity Theory. Texts in Theoretical Computer Science. An EATCS Series. Springer, Berlin (2006)zbMATHGoogle Scholar
  14. 14.
    Huang, J., Darwiche, A.: DPLL with a trace: from SAT to knowledge compilation. In: Kaelbling, L.P., Saffiotti, A. (eds.) Proceedings of the Nineteenth International Joint Conference on Artificial Intelligence, IJCAI 2005, Edinburgh, Scotland, UK, 30 July–5 August 2005, pp. 156–162. Professional Book Center (2005)Google Scholar
  15. 15.
    Koriche, F., Lagniez, J.-M., Marquis, P., Thomas, S.: Compilation, knowledge compilation for model counting : affine decision trees. In Rossi, R. (ed.) Proceedings of the 23rd International Joint Conference on Artificial Intelligence, IJCAI 2013, Beijing, China, 3–9 August 2013. IJCAI/AAAI (2013)Google Scholar
  16. 16.
    Lampis, M.: Algorithmic meta-theorems for restrictions of treewidth. In: de Berg, M., Meyer, U. (eds.) ESA 2010, Part I. LNCS, vol. 6346, pp. 549–560. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  17. 17.
    Muise, C., McIlraith, S.A., Beck, J.C., Hsu, E.I.: Dsharp: Fast d-DNNF Compilation with \(\sf sharpSAT\). In: Kosseim, L., Inkpen, D. (eds.) Canadian AI 2012. LNCS, vol. 7310, pp. 356–361. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  18. 18.
    Ordyniak, S., Paulusma, D., Szeider, S.: Satisfiability of acyclic and almost acyclic CNF formulas. Theor. Comput. Sci. 481, 85–99 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Paulusma, D., Slivovsky, F., Szeider, S.: Model counting for CNF formulas of bounded modular treewidth. In: Portier, N., Wilke, T. (eds.) 30th International Symposium on Theoretical Aspects of Computer Science, STACS 2013, Kiel, Germany, 27 February–2 March 2013. LIPIcs, vol. 20, pp. 55–66. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik (2013)Google Scholar
  20. 20.
    Razgon, I.: No small nondeterministic read-once branching programs for CNFs of bounded treewidth. In: Cygan, M., Heggernes, P. (eds.) IPEC 2014. LNCS, vol. 8894, pp. 319–331. Springer, Heidelberg (2014)Google Scholar
  21. 21.
    Razgon, I.: On OBDDs for CNFs of bounded treewidth. In: Baral, C., De Giacomo, G., Eiter, T. (eds.) Principles of Knowledge Representation and Reasoning: Proceedings of the Fourteenth International Conference, KR 2014, Vienna, Austria, 20–24 July 2014. AAAI Press (2014)Google Scholar
  22. 22.
    Sæther, S.H., Telle, J.A.: Between treewidth and clique-width. In: Kratsch, D., Todinca, I. (eds.) WG 2014. LNCS, vol. 8747, pp. 396–407. Springer, Heidelberg (2014)Google Scholar
  23. 23.
    Sang, T., Bacchus, F., Beame, P., Kautz, H.A., Pitassi, T.: Combining component caching and clause learning for effective model counting. In: Proceedings of the Seventh International Conference on Theory and Applications of Satisfiability Testing, SAT 2004, Vancouver, BC, Canada, 10–13 May 2004Google Scholar
  24. 24.
    Thurley, M.: sharpSAT – counting models with advanced component caching and implicit BCP. In: Biere, A., Gomes, C.P. (eds.) SAT 2006. LNCS, vol. 4121, pp. 424–429. Springer, Heidelberg (2006)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.CNRS, CRIL UMR 8188LensFrance

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