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New Width Parameters for Model Counting

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Theory and Applications of Satisfiability Testing – SAT 2017 (SAT 2017)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10491))

Abstract

We study the parameterized complexity of the propositional model counting problem #SAT for CNF formulas. As the parameter we consider the treewidth of the following two graphs associated with CNF formulas: the consensus graph and the conflict graph. Both graphs have as vertices the clauses of the formula; in the consensus graph two clauses are adjacent if they do not contain a complementary pair of literals, while in the conflict graph two clauses are adjacent if they do contain a complementary pair of literals. We show that #SAT is fixed-parameter tractable for the treewidth of the consensus graph but W[1]-hard for the treewidth of the conflict graph. We also compare the new parameters with known parameters under which #SAT is fixed-parameter tractable.

Supported by the Austrian Science Fund (FWF), project P26696. Robert Ganian is also affiliated with FI MU, Brno, Czech Republic.

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References

  1. Bacchus, F., Dalmao, S., Pitassi, T.: Algorithms and complexity results for #SAT and Bayesian inference. In: 44th Annual IEEE Symposium on Foundations of Computer Science (FOCS 2003), pp. 340–351 (2003)

    Google Scholar 

  2. Bodlaender, H.L.: A tourist guide through treewidth. Acta Cybernetica 11, 1–21 (1993)

    MathSciNet  MATH  Google Scholar 

  3. Bodlaender, H.L.: A linear-time algorithm for finding tree-decompositions of small treewidth. SIAM J. Comput. 25(6), 1305–1317 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bodlaender, H.L., Drange, P.G., Dregi, M.S., Fomin, F.V., Lokshtanov, D., Pilipczuk, M.: A c\({}^{\text{ k }}\) n 5-approximation algorithm for treewidth. SIAM J. Comput. 45(2), 317–378 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bodlaender, H.L., Kloks, T.: Efficient and constructive algorithms for the pathwidth and treewidth of graphs. J. Algorithms 21(2), 358–402 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  6. 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)

    Article  MathSciNet  MATH  Google Scholar 

  7. Diestel, R.: Graph Theory. Graduate Texts in Mathematics, vol. 173, 4th edn. Springer, New York (2010)

    Book  MATH  Google Scholar 

  8. Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity. Texts in Computer Science. Springer, London (2013)

    Book  MATH  Google Scholar 

  9. Fürer, M.: Faster integer multiplication. SIAM J. Comput. 39(3), 979–1005 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  10. Galesi, N., Kullmann, O.: Polynomial time SAT decision, hypergraph transversals and the hermitian rank. In: Hoos, H.H., Mitchell, D.G. (eds.) SAT 2004. LNCS, vol. 3542, pp. 89–104. Springer, Heidelberg (2005). doi:10.1007/11527695_8

    Chapter  Google Scholar 

  11. Ganian, R., Hlinený, P., Obdrzálek, J.: Better algorithms for satisfiability problems for formulas of bounded rank-width. Fund. Inform. 123(1), 59–76 (2013)

    MathSciNet  MATH  Google Scholar 

  12. Ganian, R., Szeider, S.: Community structure inspired algorithms for SAT and #SAT. In: Heule, M., Weaver, S. (eds.) SAT 2015. LNCS, vol. 9340, pp. 223–237. Springer, Cham (2015). doi:10.1007/978-3-319-24318-4_17

    Chapter  Google Scholar 

  13. Iwama, K.: CNF-satisfiability test by counting and polynomial average time. SIAM J. Comput. 18(2), 385–391 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  14. Kleine Büning, H., Kullmann, O.: Minimal unsatisfiability and autarkies. In: Biere, A., Heule, M.J.H., van Maaren, H., Walsh, T. (eds.) Handbook of Satisfiability, Frontiers in Artificial Intelligence and Applications, vol. 185, chap. 11, pp. 339–401. IOS Press (2009)

    Google Scholar 

  15. Kleine Büning, H., Zhao, X.: Satisfiable formulas closed under replacement. In: Kautz, H., Selman, B. (eds.) Proceedings for the Workshop on Theory and Applications of Satisfiability. Electronic Notes in Discrete Mathematics, vol. 9. Elsevier Science Publishers, North-Holland (2001)

    Google Scholar 

  16. Kloks, T.: Treewidth: Computations and Approximations. Springer, Berlin (1994)

    Book  MATH  Google Scholar 

  17. Knuth, D.E.: How fast can we multiply? In: The Art of Computer Programming. Seminumerical Algorithms, 3rd edn., vol. 2, chap. 4.3.3, pp. 294–318. Addison-Wesley (1998)

    Google Scholar 

  18. Kullmann, O.: The combinatorics of conflicts between clauses. In: Giunchiglia, E., Tacchella, A. (eds.) SAT 2003. LNCS, vol. 2919, pp. 426–440. Springer, Heidelberg (2004). doi:10.1007/978-3-540-24605-3_32

    Chapter  Google Scholar 

  19. Niedermeier, R.: Invitation to Fixed-Parameter Algorithms. Oxford Lecture Series in Mathematics and its Applications. Oxford University Press, Oxford (2006)

    Book  MATH  Google Scholar 

  20. Nishimura, N., Ragde, P., Szeider, S.: Solving #SAT using vertex covers. Acta Informatica 44(7–8), 509–523 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  21. Ordyniak, S., Paulusma, D., Szeider, S.: Satisfiability of acyclic and almost acyclic CNF formulas. Theor. Comput. Sci. 481, 85–99 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  22. Robertson, N., Seymour, P.D.: Graph minors. II. Algorithmic aspects of tree-width. J. Algorithms 7(3), 309–322 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  23. Roth, D.: On the hardness of approximate reasoning. Artif. Intell. 82(1–2), 273–302 (1996)

    Article  MathSciNet  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  25. Samer, M., Szeider, S.: Fixed-parameter tractability. In: Biere, A., Heule, M., van Maaren, H., Walsh, T. (eds.) Handbook of Satisfiability, chap. 13, pp. 425–454. IOS Press (2009)

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  27. Scheder, D., Zumstein, P.: How many conflicts does it need to be unsatisfiable? In: Kleine Büning, H., Zhao, X. (eds.) SAT 2008. LNCS, vol. 4996, pp. 246–256. Springer, Heidelberg (2008). doi:10.1007/978-3-540-79719-7_23

    Chapter  Google Scholar 

  28. Szeider, S.: On fixed-parameter tractable parameterizations of SAT. In: Giunchiglia, E., Tacchella, A. (eds.) SAT 2003. LNCS, vol. 2919, pp. 188–202. Springer, Heidelberg (2004). doi:10.1007/978-3-540-24605-3_15

    Chapter  Google Scholar 

  29. Valiant, L.G.: The complexity of computing the permanent. Theoret. Comput. Sci. 8(2), 189–201 (1979)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Algorithms and Complexity Group, TU Wien, Vienna, Austria

    Robert Ganian & Stefan Szeider

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  1. Robert Ganian
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  2. Stefan Szeider
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Correspondence to Robert Ganian .

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Editors and Affiliations

  1. UNSW Sydney, Sydney, New South Wales, Australia

    Serge Gaspers

  2. University of New South Wales , Sydney, New South Wales, Australia

    Toby Walsh

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Ganian, R., Szeider, S. (2017). New Width Parameters for Model Counting. 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_3

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  • DOI: https://doi.org/10.1007/978-3-319-66263-3_3

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