QBF as an Alternative to Courcelle’s Theorem

  • Michael Lampis
  • Stefan Mengel
  • Valia MitsouEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10929)


We propose reductions to quantified Boolean formulas (QBF) as a new approach to showing fixed-parameter linear algorithms for problems parameterized by treewidth. We demonstrate the feasibility of this approach by giving new algorithms for several well-known problems from artificial intelligence that are in general complete for the second level of the polynomial hierarchy. By reduction from QBF we show that all resulting algorithms are essentially optimal in their dependence on the treewidth. Most of the problems that we consider were already known to be fixed-parameter linear by using Courcelle’s Theorem or dynamic programming, but we argue that our approach has clear advantages over these techniques: on the one hand, in contrast to Courcelle’s Theorem, we get concrete and tight guarantees for the runtime dependence on the treewidth. On the other hand, we avoid tedious dynamic programming and, after showing some normalization results for CNF-formulas, our upper bounds often boil down to a few lines.


Quantified Boolean Formulas (QBF) Polynomial Hierarchy Runtime Dependencies Minimal Unsatisfiable Set Incidence Treewidth 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



Most of the research in this paper was performed during a stay of the first and third authors at CRIL that was financed by the project PEPS INS2I 2017 CODA. The second author is thankful for many valuable discussions with members of CRIL, in particular Jean-Marie Lagniez, Emmanuel Lonca and Pierre Marquis, on the topic of this article.

Moreover, the authors would like to thank the anonymous reviewers whose numerous helpful remarks allowed to improve the presentation of the paper.


  1. 1.
    Arieli, O., Caminada, M.W.A.: A general QBF-based formalization of abstract argumentation theory. In: Verheij, B., Szeider, S., Woltran, S. (eds.) Computational Models of Argument, COMMA 2012, pp. 105–116 (2012)Google Scholar
  2. 2.
    Arnborg, S., Proskurowski, A.: Linear time algorithms for NP-hard problems restricted to partial k-trees. Discrete Appl. Math. 23(1), 11–24 (1989)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Atserias, A., Oliva, S.: Bounded-width QBF is PSPACE-complete. J. Comput. Syst. Sci. 80(7), 1415–1429 (2014)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Bodlaender, H.L.: A linear-time algorithm for finding tree-decompositions of small treewidth. SIAM J. Comput. 25(6), 1305–1317 (1996)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Bodlaender, H.L., Drange, P.G., Dregi, M.S., Fomin, F.V., Lokshtanov, D., Pilipczuk, M.: An o(c\({}^{\wedge }\)k n) 5-approximation algorithm for treewidth. In: 54th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2013, pp. 499–508 (2013)Google Scholar
  6. 6.
    Cadoli, M., Lenzerini, M.: The complexity of propositional closed world reasoning and circumscription. J. Comput. Syst. Sci. 48(2), 255–310 (1994)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Chen, H.: Quantified constraint satisfaction and bounded treewidth. In: de Mántaras, R.L., Saitta, L. (eds.) Proceedings of the 16th European Conference on Artificial Intelligence, ECAI 2004, pp. 161–165 (2004)Google Scholar
  8. 8.
    Courcelle, B.: The monadic second-order logic of graphs. I. Recognizable sets of finite graphs. Inf. Comput. 85(1), 12–75 (1990)MathSciNetCrossRefGoogle Scholar
  9. 9.
    DIMACS: Satisfiability: Suggested Format. DIMACS Challenge. DIMACS (1993)Google Scholar
  10. 10.
    Dung, P.M.: On the acceptability of arguments and its fundamental role in nonmonotonic reasoning, logic programming and n-person games. Artif. Intell. 77(2), 321–358 (1995)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Dunne, P.E.: Computational properties of argument systems satisfying graph-theoretic constraints. Artif. Intell. 171(10–15), 701–729 (2007)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Dunne, P.E., Bench-Capon, T.J.M.: Coherence in finite argument systems. Artif. Intell. 141(1/2), 187–203 (2002)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Dvorák, W., Pichler, R., Woltran, S.: Towards fixed-parameter tractable algorithms for abstract argumentation. Artif. Intell. 186, 1–37 (2012)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Egly, U., Woltran, S.: Reasoning in argumentation frameworks using quantified boolean formulas. In: Dunne, P.E., Bench-Capon, T.J.M. (eds.) Computational Models of Argument, COMMA 2006, pp. 133–144 (2006)Google Scholar
  15. 15.
    Eiben, E., Ganian, R., Ordyniak, S.: Using decomposition-parameters for QBF: mind the prefix! In: Schuurmans, D., Wellman, M.P. (eds.) Proceedings of the Thirtieth AAAI Conference on Artificial Intelligence, pp. 964–970 (2016)Google Scholar
  16. 16.
    Eiben, E., Ganian, R., Ordyniak, S.: Small resolution proofs for QBF using dependency treewidth. CoRR, abs/1711.02120 (2017)Google Scholar
  17. 17.
    Eiter, T., Gottlob, G.: Propositional circumscription and extended closed-world reasoning are \(\varPi ^{p}_2\)-complete. Theor. Comput. Sci. 114(2), 231–245 (1993)CrossRefGoogle Scholar
  18. 18.
    Eiter, T., Gottlob, G.: The complexity of logic-based abduction. J. ACM 42(1), 3–42 (1995)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Fischer, E., Makowsky, J.A., Ravve, E.V.: Counting truth assignments of formulas of bounded tree-width or clique-width. Discrete Appl. Math. 156(4), 511–529 (2008)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Frick, M., Grohe, M.: The complexity of first-order and monadic second-order logic revisited. Ann. Pure Appl. Logic 130(1–3), 3–31 (2004)MathSciNetCrossRefGoogle Scholar
  21. 21.
    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)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Ignatiev, A., Previti, A., Liffiton, M., Marques-Silva, J.: Smallest MUS extraction with minimal hitting set dualization. In: Pesant, G. (ed.) CP 2015. LNCS, vol. 9255, pp. 173–182. Springer, Cham (2015). Scholar
  23. 23.
    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). Scholar
  24. 24.
    Janota, M., Marques-Silva, J.: On deciding MUS membership with QBF. In: Lee, J. (ed.) CP 2011. LNCS, vol. 6876, pp. 414–428. Springer, Heidelberg (2011). Scholar
  25. 25.
    Kloks, T. (ed.): Treewidth: Computations and Approximations. LNCS, vol. 842. Springer, Heidelberg (1994). Scholar
  26. 26.
    Lagniez, J.-M., Biere, A.: Factoring out assumptions to speed up MUS extraction. In: Järvisalo, M., Van Gelder, A. (eds.) SAT 2013. LNCS, vol. 7962, pp. 276–292. Springer, Heidelberg (2013). Scholar
  27. 27.
    Lagniez, J.-M., Lonca, E., Mailly, J.-G.: CoQuiAAS: a constraint-based quick abstract argumentation solver. In: 27th IEEE International Conference on Tools with Artificial Intelligence, ICTAI 2015, pp. 928–935 (2015)Google Scholar
  28. 28.
    Lampis, M., Mitsou, V.: Treewidth with a quantifier alternation revisited (2017)Google Scholar
  29. 29.
    Langer, A., Reidl, F., Rossmanith, P., Sikdar, S.: Practical algorithms for MSO model-checking on tree-decomposable graphs. Comput. Sci. Rev. 13–14, 39–74 (2014)CrossRefGoogle Scholar
  30. 30.
    Liberatore, P.: Redundancy in logic I: CNF propositional formulae. Artif. Intell. 163(2), 203–232 (2005)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Lifschitz, V.: Handbook of Logic in Artificial Intelligence and Logic Programming, vol. 3. Chapter Circumscription, pp. 297–352. Oxford University Press Inc., New York (1994)Google Scholar
  32. 32.
    Lokshtanov, D., Marx, D., Saurabh, S.: Lower bounds based on the exponential time hypothesis. Bull. EATCS 105, 41–72 (2011)MathSciNetzbMATHGoogle Scholar
  33. 33.
    Marx, D., Mitsou, V.: Double-exponential and triple-exponential bounds for choosability problems parameterized by treewidth. In: 43rd International Colloquium on Automata, Languages, and Programming, ICALP 2016, pp. 28:1–28:15 (2016)Google Scholar
  34. 34.
    McCarthy, J.: Applications of circumscription to formalizing common-sense knowledge. Artif. Intell. 28(1), 89–116 (1986)MathSciNetCrossRefGoogle Scholar
  35. 35.
    Pan, G., Vardi, M.Y.: Fixed-parameter hierarchies inside PSPACE. In: 21st IEEE Symposium on Logic in Computer Science, LICS 2006, pp. 27–36 (2006)Google Scholar
  36. 36.
    Samer, M., Szeider, S.: Algorithms for propositional model counting. J. Discrete Algorithms 8(1), 50–64 (2010)MathSciNetCrossRefGoogle Scholar
  37. 37.
    Schaefer, M., Umans, C.: Completeness in the polynomial-time hierarchy: a compendium. SIGACT News 33(3), 32–49 (2002)CrossRefGoogle Scholar
  38. 38.
    Marques-Silva, J., Lynce, I.: On improving MUS extraction algorithms. In: Sakallah, K.A., Simon, L. (eds.) SAT 2011. LNCS, vol. 6695, pp. 159–173. Springer, Heidelberg (2011). Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Université Paris-Dauphine, PSL Research University, CNRS, UMR 7243ParisFrance
  2. 2.CNRS, CRIL UMR 8188ParisFrance
  3. 3.Université Paris-Diderot, IRIF, CNRS, UMR 8243ParisFrance

Personalised recommendations