Community Structure Inspired Algorithms for SAT and #SAT

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9340)

Abstract

We introduce h-modularity, a structural parameter of CNF formulas, and present algorithms that render the decision problem SAT and the model counting problem #SAT fixed-parameter tractable when parameterized by h-modularity. The new parameter is defined in terms of a partition of clauses of the given CNF formula into strongly interconnected communities which are sparsely interconnected with each other. Each community forms a hitting formula, whereas the interconnections between communities form a graph of small treewidth. Our algorithms first identify the community structure and then use them for an efficient solution of SAT and #SAT, respectively. We further show that h-modularity is incomparable with known parameters under which SAT or #SAT is fixed-parameter tractable.

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References

  1. 1.
    Ansótegui, C., Bonet, M.L., Giráldez-Cru, J., Levy, J.: The fractal dimension of SAT formulas. In: Demri, S., Kapur, D., Weidenbach, C. (eds.) IJCAR 2014. LNCS, vol. 8562, pp. 107–121. Springer, Heidelberg (2014) Google Scholar
  2. 2.
    Bacchus, F., Dalmao, S., Pitassi, T.: Solving #SAT and Bayesian inference with backtracking search. J. Artif. Intell. Res. 34, 391–442 (2009)MathSciNetMATHGoogle Scholar
  3. 3.
    Bodlaender, H.L.: A linear-time algorithm for finding tree-decompositions of small treewidth. SIAM J. Comput. 25(6), 1305–1317 (1996)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    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)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Crama, Y., Hammer, P.L.: Boolean functions. Encyclopedia of Mathematics and its Applications, vol. 142. Cambridge University Press, Cambridge (2011). Theory, algorithms, and applications CrossRefMATHGoogle Scholar
  6. 6.
    Diestel, R.: Graph Theory. Graduate Texts in Mathematics, vol. 173, 4th edn. Springer Verlag, New York (2010)CrossRefMATHGoogle Scholar
  7. 7.
    Downey, R.G., Fellows, M.R.: Parameterized Complexity. Monographs in Computer Science. Springer, New York (1999)CrossRefMATHGoogle Scholar
  8. 8.
    Fleischner, H., Kullmann, O., Szeider, S.: Polynomial-time recognition of minimal unsatisfiable formulas with fixed clause-variable difference. Theoretical Computer Science 289(1), 503–516 (2002)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Ganian, R., Hlinený, P., Obdrzálek, J.: Better algorithms for satisfiability problems for formulas of bounded rank-width. Fund. Inform. 123(1), 59–76 (2013)MathSciNetMATHGoogle Scholar
  10. 10.
    Iwama, K.: CNF-satisfiability test by counting and polynomial average time. SIAM J. Comput. 18(2), 385–391 (1989)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Büning, H.K., 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, chapter 11, pp. 339–401. IOS Press (2009)Google Scholar
  12. 12.
    Büning, H.K., 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
  13. 13.
    Büning, K.H., Zhao, X.: On the structure of some classes of minimal unsatisfiable formulas. Discr. Appl. Math. 130(2), 185–207 (2003)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Kloks, T.: Treewidth: Computations and Approximations. Springer Verlag, Berlin (1994)CrossRefMATHGoogle Scholar
  15. 15.
    Kullmann, O.: Lean clause-sets: Generalizations of minimally unsatisfiable clause-sets. Discr. Appl. Math. 130(2), 209–249 (2003)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Marx, D.: Parameterized complexity and approximation algorithms. The Computer Journal 51(1), 60–78 (2008)CrossRefGoogle Scholar
  17. 17.
    Newman, M.E.J.: The structure and function of complex networks. SIAM Review 45(2), 167–256 (2003)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Newman, M.E.J.: Modularity and community structure in networks. Proceedings of the National Academy of Sciences 103(23), 8577–8582 (2006)CrossRefGoogle Scholar
  19. 19.
    Newman, M.E.J., Girvan, M.: Finding and evaluating community structure in networks. Phys. Rev. E 69(2), 026113 (2004)CrossRefGoogle Scholar
  20. 20.
    Newsham, Z., Ganesh, V., Fischmeister, S., Audemard, G., Simon, L.: Impact of community structure on SAT solver performance. In: Sinz, C., Egly, U. (eds.) SAT 2014. LNCS, vol. 8561, pp. 252–268. Springer, Heidelberg (2014) Google Scholar
  21. 21.
    Niedermeier, R.: Invitation to Fixed-Parameter Algorithms. Oxford Lecture Series in Mathematics and its Applications. Oxford University Press, Oxford (2006) CrossRefMATHGoogle Scholar
  22. 22.
    Nishimura, N., Ragde, P., Szeider, S.: Solving #SAT using vertex covers. Acta Informatica 44(7–8), 509–523 (2007)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Robertson, N., Seymour, P.D.: Graph minors. II. Algorithmic aspects of tree-width. J. Algorithms 7(3), 309–322 (1986)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Rose, D.J.: On simple characterizations of \(k\)-trees. Discrete Math. 7, 317–322 (1974)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Samer, M., Szeider, S.: Fixed-parameter tractability. In: Biere, A., Heule, M., van Maaren, H., Walsh, T. (eds.) Handbook of Satisfiability, chapter 13, pp. 425–454. IOS Press (2009)Google Scholar
  26. 26.
    Samer, M., Szeider, S.: Algorithms for propositional model counting. J. Discrete Algorithms 8(1), 50–64 (2010)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Szeider, S.: Minimal unsatisfiable formulas with bounded clause-variable difference are fixed-parameter tractable. J. of Computer and System Sciences 69(4), 656–674 (2004)MathSciNetCrossRefMATHGoogle Scholar
  28. 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) CrossRefGoogle Scholar
  29. 29.
    Vardi, M.Y.: Boolean satisfiability: theory and engineering. Communications of the ACM 57(3), 5 (2014)CrossRefGoogle Scholar
  30. 30.
    Živný, S.: The Complexity of Valued Constraint Satisfaction Problems. Cognitive Technologies. Springer (2012)Google Scholar
  31. 31.
    Zhang, W., Pan, G., Wu, Z., Li, S.: Online community detection for large complex networks. In: Rossi, F. (eds.) Proceedings of the 23rd International Joint Conference on Artificial Intelligence, IJCAI 2013, Beijing, China, August 3–9, 2013. IJCAI/AAAI (2013)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Algorithms and Complexity GroupTU WienViennaAustria

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