Abstract
We propose an exact algorithm for counting the models of propositional formulas in conjunctive normal form (CNF). Our algorithm is based on the detection of strong backdoor sets of bounded size; each instantiation of the variables of a strong backdoor set puts the given formula into a class of formulas for which models can be counted in polynomial time. For the backdoor set detection we utilize an efficient vertex cover algorithm applied to a certain “obstruction graph” that we associate with the given formula. This approach gives rise to a new hardness index for formulas, the clustering-width. Our algorithm runs in uniform polynomial time on formulas with bounded clustering-width.
It is known that the number of models of formulas with bounded clique-width, bounded treewidth, or bounded branchwidth can be computed in polynomial time; these graph parameters are applied to formulas via certain (hyper)graphs associated with formulas. We show that clustering-width and the other parameters mentioned are incomparable: there are formulas with bounded clustering-width and arbitrarily large clique-width, treewidth, and branchwidth. Conversely, there are formulas with arbitrarily large clustering-width and bounded clique-width, treewidth, and branchwidth.
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
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)
Bodlaender, H.L.: A partial k-arboretum of graphs with bounded treewidth. Theoret. Comput. Sci. 209(1-2), 1–45 (1998)
Chen, J., Kanj, I.A., Xia, G.: Simplicity is beauty: Improved upper bounds for vertex cover. Technical Report TR05-008, DePaul University, Chicago IL (2005)
Courcelle, B., Makowsky, J.A., Rotics, U.: Linear time solvable optimization problems on graphs of bounded clique-width. Theory Comput. Syst. 33(2), 125–150 (2000)
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)
Courcelle, B., Olariu, S.: Upper bounds to the clique width of graphs. Discr. Appl. Math. 101(1-3), 77–114 (2000)
Downey, R.G., Fellows, M.R.: Parameterized Complexity. In: Monographs in Computer Science, Springer, Heidelberg (1999)
Fischer, E., Makowsky, J.A., Ravve, E.R.: Counting truth assignments of formulas of bounded tree-width or clique-width. Discr. Appl. Math. (to appear)
Flum, J., Grohe, M.: The parameterized complexity of counting problems. SIAM J. Comput. 33(4), 892–922 (2004)
Golumbic, M.C., Rotics, U.: On the clique-width of some perfect graph classes. WG 1999 11(3), 423–443 (2000), Selected papers from the Workshop on Graph-Theoretical Aspects of Computer Science (WG 1999), Part 1 (Ascona)
Gramm, J., Guo, J., Hüffner, F., Niedermeier, R.: Graph-modeled data clustering: fixed-parameter algorithms for clique generation. Theory Comput. Syst. 38(4), 373–392 (2005)
Gottlob, G., Scarcello, F., Sideri, M.: Fixed-parameter complexity in AI and nonmonotonic reasoning. Artificial Intelligence 138(1-2), 55–86 (2002)
Gottlob, G., Szeider, S.: Fixed-parameter algorithms for artificial intelligence, constraint satisfaction, and database problems (April 2006) (submitted)
Interian, Y.: Backdoor sets for random 3-SAT. In: Informal Proceedings of SAT 2003, pp. 231–238 (2003)
Iwama, K.: CNF-satisfiability test by counting and polynomial average time. SIAM J. Comput. 18(2), 385–391 (1989)
Kilby, P., Slaney, J.K., Thiébaux, S., Walsh, T.: Backbones and backdoors in satisfiability. In: Proceedings, The Twentieth National Conference on Artificial Intelligence and the Seventeenth Innovative Applications of Artificial Intelligence Conference (AAAI 2005), pp. 1368–1373 (2005)
Kleine Büning, H., Zhao, X.: Satisfiable formulas closed under replacement. In: Proceedings for the Workshop on Theory and Applications of Satisfiability. Electronic Notes in Discrete Mathematics, vol. 9, Elsevier Science Publishers, North-Holland (2001)
Lynce, I., Marques-Silva, J.P.: Hidden structure in unsatisfiable random 3-SAT: An empirical study. In: 16th IEEE International Conference on Tools with Artificial Intelligence (ICTAI 2004), pp. 246–251. IEEE Computer Society, Los Alamitos (2004)
Niedermeier, R.: Invitation to Fixed-Parameter Algorithms. In: Oxford Lecture Series in Mathematics and Its Applications, Oxford University Press, Oxford (2006)
Nishimura, N., Ragde, P., Szeider, S.: Detecting backdoor sets with respect to Horn and binary clauses. In: Informal Proceedings of SAT 2004, pp. 96–103 (2004)
Oum, S., Seymour, P.: Approximating clique-width and branch-width. J. Combin. Theory, Ser. B (to appear)
Robertson, N., Seymour, P.D.: Graph minors. X. Obstructions to tree-decomposition. J. Combin. Theory Ser. B 52(2), 153–190 (1991)
Roth, D.: On the hardness of approximate reasoning. Artificial Intelligence 82(1-2), 273–302 (1996)
Ruan, Y., Kautz, H.A., Horvitz, E.: The backdoor key: A path to understanding problem hardness. In: Proceedings of the 19th National Conference on Artificial Intelligence, 16th Conference on Innovative Applications of Artificial Intelligence, pp. 124–130. AAAI Press / The MIT Press (2004)
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)
Szeider, S.: Backdoor sets for DLL subsolvers. Journal of Automated Reasoning (in press, 2005)
Valiant, L.G.: The complexity of computing the permanent. Theoret. Comput. Sci. 8(2), 189–201 (1979)
Williams, R., Gomes, C., Selman, B.: On the connections between backdoors, restarts, and heavy-tailedness in combinatorial search. In: Informal Proceedings of SAT 2003, pp. 222–230 (2003)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2006 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Nishimura, N., Ragde, P., Szeider, S. (2006). Solving #SAT Using Vertex Covers. In: Biere, A., Gomes, C.P. (eds) Theory and Applications of Satisfiability Testing - SAT 2006. SAT 2006. Lecture Notes in Computer Science, vol 4121. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11814948_36
Download citation
DOI: https://doi.org/10.1007/11814948_36
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-37206-6
Online ISBN: 978-3-540-37207-3
eBook Packages: Computer ScienceComputer Science (R0)