Skip to main content
SpringerLink
Log in
Book cover

International Conference on Theory and Applications of Satisfiability Testing

SAT 2019: Theory and Applications of Satisfiability Testing – SAT 2019 pp 222–238Cite as

Revisiting Graph Width Measures for CNF-Encodings

  • Conference paper
  • First Online:

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 11628))

Abstract

We consider bounded width CNF-formulas where the width is measured by popular graph width measures on graphs associated to CNF-formulas. Such restricted graph classes, in particular those of bounded treewidth, have been extensively studied for their uses in the design of algorithms for various computational problems on CNF-formulas. Here we consider the expressivity of these formulas in the model of clausal encodings with auxiliary variables. We first show that bounding the width for many of the measures from the literature leads to a dramatic loss of expressivity, restricting the formulas to such of low communication complexity. We then show that the width of optimal encodings with respect to different measures is strongly linked: there are two classes of width measures, one containing primal treewidth and the other incidence cliquewidth, such that in each class the width of optimal encodings only differs by constant factors. Moreover, between the two classes the width differs at most by a factor logarithmic in the number of variables. Both these results are in stark contrast to the setting without auxiliary variables where all width measures we consider here differ by more than constant factors and in many cases even by linear factors.

This is a preview of subscription content, log in via an institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

References

  1. Amarilli, A., Capelli, F., Monet, M., Senellart, P.: Connecting knowledge compilation classes and width parameters CoRR, abs/1811.02944 (2018)

    Google Scholar 

  2. Amarilli, A., Monet, M., Senellart, P.: Connecting width and structure in knowledge compilation. In: 21st International Conference on Database Theory, ICDT, Vienna, Austria, 26–29 March 2018 (2018)

    Google Scholar 

  3. Biere, A., Heule, M., van Maaren, H., Walsh, T. (eds.): Handbook of Satisfiability, volume 185 of Frontiers in Artificial Intelligence and Applications. IOS Press (2009)

    Google Scholar 

  4. 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, Cham (2015). https://doi.org/10.1007/978-3-319-24318-4_15

    Chapter  Google Scholar 

  5. Bova, S., Capelli, F., Mengel, S., Slivovsky, F.: Knowledge compilation meets communication complexity. In: Proceedings of the Twenty-Fifth International Joint Conference on Artificial Intelligence, IJCAI 2016 (2016)

    Google Scholar 

  6. Bova, S., Szeider, S.: Circuit treewidth, sentential decision, and query compilation (2017)

    Google Scholar 

  7. Brault-Baron, J., Capelli, F., Mengel, S.: Understanding model counting for \(\beta \)-acyclic CNF-formulas CoRR, abs/1405.6043 (2014)

    Google Scholar 

  8. Briquel, I., Koiran, P., Meer, K.: On the expressive power of CNF formulas of bounded tree- and clique-width. Discrete Appl. Math. 159(1), 1–14 (2011)

    Article  MathSciNet  Google Scholar 

  9. Capelli, F., Mengel, S.: Tractable QBF by knowledge compilation. In: 36th International Symposium on Theoretical Aspects of Computer Science, STACS 2019, vol. 126, pp. 18:1–18:16 (2019)

    Google Scholar 

  10. Chen, H.: Quantified constraint satisfaction and bounded treewidth. In: Proceedings of the 16th Eureopean Conference on Artificial Intelligence, ECAI 2004 (2004)

    Google Scholar 

  11. Darwiche, A., Marquis, P.: A knowledge compilation map. J. Artif. Intell. Res. 17, 229–264 (2002)

    Article  MathSciNet  Google Scholar 

  12. Diestel, R.: Graph Theory. Volume 173 of Graduate texts in Mathematics, 4th edn. Springer, Heidelberg (2012)

    MATH  Google Scholar 

  13. Fischer, E., Makowsky, J., Ravve, E.: Counting truth assignments of formulas of bounded tree-width or clique-width. Discrete Appl. Math. 156(4), 511–529 (2008)

    Article  MathSciNet  Google Scholar 

  14. Ganian, R., Szeider, S.: New width parameters for model counting. In: Gaspers, S., Walsh, T. (eds.) SAT 2017. LNCS, vol. 10491, pp. 38–52. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-66263-3_3

    Chapter  Google Scholar 

  15. Gaspers, S., Szeider, S.: Strong backdoors to bounded treewidth SAT. In: 54th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2013 (2013)

    Google Scholar 

  16. Gent, I.P., Nightingale, P.: A new encoding of AllDifferent into SAT. In: International Workshop on Modelling and Reformulating Constraint Satisfaction Problems (2004)

    Google Scholar 

  17. Giunchiglia, E., Maratea, E., Tacchella, A.: Dependent and independent variables in propositional satisfiability. In: Logics in Artificial Intelligence. JELIA 2002 (2002)

    Google Scholar 

  18. Golumbic, M.C., Gurvich, V.: Read-once functions. In: Boolean Functions: Theory, Algorithms and Applications, pp. 519–560 (2011)

    Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  20. Jakl, M., Pichler, R., Woltran, S.: Answer-set programming with bounded treewidth. In: Proceedings of the 21st International Joint Conference on Artificial Intelligence. IJCAI 2009 (2009)

    Google Scholar 

  21. Krause, M.: Exponential lower bounds on the complexity of local and real-time branching programs. Elektronische Informationsverarbeitung und Kybernetik 24(3), 99–110 (1988)

    MathSciNet  MATH  Google Scholar 

  22. Kushilevitz, E., Nisan, N.: Communication Complexity. Cambridge University Press, Cambridge (1997)

    MATH  Google Scholar 

  23. Lampis, M., Mengel, S., Mitsou, V.: QBF as an alternative to Courcelle’s theorem. In: Beyersdorff, O., Wintersteiger, C.M. (eds.) SAT 2018. LNCS, vol. 10929, pp. 235–252. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-94144-8_15

    Chapter  Google Scholar 

  24. Paulusma, D., Slivovsky, F., Szeider, S.: Model counting for CNF formulas of bounded modular treewidth. In: 30th International Symposium on Theoretical Aspects of Computer Science, STACS 2013 (2013)

    Google Scholar 

  25. Paulusma, D., Slivovsky, F., Szeider, S.: Model counting for CNF formulas of bounded modular treewidth. Algorithmica 76(1), 168–194 (2016)

    Article  MathSciNet  Google Scholar 

  26. Pipatsrisawat, K., Darwiche, A.: New compilation languages based on structured decomposability. In: Proceedings of the Twenty-Third AAAI Conference on Artificial Intelligence, AAAI 2008 (2008)

    Google Scholar 

  27. Pipatsrisawat, T., Darwiche, A.: A lower bound on the size of decomposable negation normal form. In: Proceedings of the Twenty-Fourth AAAI Conference on Artificial Intelligence, AAAI 2010 (2010)

    Google Scholar 

  28. Sæther, S.H., Telle, J.A., Vatshelle, M.: Solving #SAT and MAXSAT by dynamic programming. J. Artif. Intell. Res. 54, 59–82 (2015)

    Article  MathSciNet  Google Scholar 

  29. Samer, M., Szeider, S.: Algorithms for propositional model counting. J. Discrete Algorithms 8(1), 50–64 (2010)

    Article  MathSciNet  Google Scholar 

  30. Samer, M., Szeider, S.: Constraint satisfaction with bounded treewidth revisited. J. Comput. Syst. Sci. 76(2), 103–114 (2010)

    Article  MathSciNet  Google Scholar 

  31. Sinz, C.: Towards an optimal CNF encoding of boolean cardinality constraints. In: 11th International Conference Principles and Practice of Constraint Programming - CP 2005. CP 2005 (2005)

    Chapter  Google Scholar 

  32. Slivovsky, F., Szeider, S.: Model counting for formulas of bounded clique-width. In: 24th International Symposium on Algorithms and Computation. ISAAC 2013 (2013)

    Chapter  Google Scholar 

  33. Vatshelle, M.: New width parameters of graphs. Ph.D. Thesis, University of Bergen (2012)

    Google Scholar 

  34. Wegener, I.: Branching Programs and Binary Decision Diagrams. SIAM (2000)

    Google Scholar 

Download references

Acknowledgements

The authors are grateful to the anonymous reviewers for their comments, which greatly helped to improve the presentation of the paper. The first author would also like to thank David Mitchell for asking the right question at the right moment. This paper grew largely out of an answer to this question.

Author information

Authors and Affiliations

  1. CRIL-CNRS UMR 8188, 62307, Lens, France

    Stefan Mengel & Romain Wallon

  2. Université d’Artois, Lens, France

    Romain Wallon

Authors
  1. Stefan Mengel
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Romain Wallon
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Romain Wallon .

Editor information

Editors and Affiliations

  1. University of Lisbon, Lisbon, Portugal

    Mikoláš Janota

  2. University of Lisbon, Lisbon, Portugal

    Inês Lynce

Rights and permissions

Reprints and permissions

Copyright information

© 2019 Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Mengel, S., Wallon, R. (2019). Revisiting Graph Width Measures for CNF-Encodings. In: Janota, M., Lynce, I. (eds) Theory and Applications of Satisfiability Testing – SAT 2019. SAT 2019. Lecture Notes in Computer Science(), vol 11628. Springer, Cham. https://doi.org/10.1007/978-3-030-24258-9_16

Download citation

  • DOI: https://doi.org/10.1007/978-3-030-24258-9_16

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-24257-2

  • Online ISBN: 978-3-030-24258-9

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation