Abstract
A backbone of a propositional CNF formula is a variable whose truth value is the same in every truth assignment that satisfies the formula. The notion of backbones for CNF formulas has been studied in various contexts. In this paper, we introduce local variants of backbones, and study the computational complexity of detecting them. In particular, we consider k- backbones, which are backbones for sub-formulas consisting of at most k clauses, and iterative k- backbones, which are backbones that result after repeated instantiations of k- backbones. We determine the parameterized complexity of deciding whether a variable is a k- backbone or an iterative k- backbone for various restricted formula classes, including Horn, definite Horn, and Krom. We also present some first empirical results regarding backbones for CNF-Satisfiability (SAT). The empirical results we obtain show that a large fraction of the backbones of structured SAT instances are local, in contrast to random instances, which appear to have few local backbones.
Access provided by Autonomous University of Puebla. Download to read the full chapter text
Chapter PDF
Similar content being viewed by others
Keywords
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.
References
Aharoni, R., Linial, N.: Minimal non-two-colorable hypergraphs and minimal unsatisfiable formulas. J. Combin. Theory Ser. A 43, 196–204 (1986)
Belov, A., Marques-Silva, J.: MUSer2: An efficient MUS extractor. J. on Satisfiability, Boolean Modeling and Computation 8(1/2), 123–128 (2012)
Buresh-Oppenheim, J., Mitchell, D.: Minimum 2CNF resolution refutations in polynomial time. In: Marques-Silva, J., Sakallah, K.A. (eds.) SAT 2007. LNCS, vol. 4501, pp. 300–313. Springer, Heidelberg (2007)
Buresh-Oppenheim, J., Mitchell, D.: Minimum witnesses for unsatisfiable 2CNFs. In: Biere, A., Gomes, C.P. (eds.) SAT 2006. LNCS, vol. 4121, pp. 42–47. Springer, Heidelberg (2006)
Darwiche, A., Marquis, P.: A knowledge compilation map. J. Artif. Intell. Res. 17, 229–264 (2002)
Dowling, W.F., Gallier, J.H.: Linear-time algorithms for testing the satisfiability of propositional horn formulae. J. Logic Programming 1(3), 267–284 (1984)
Downey, R.G., Fellows, M.R.: Parameterized Complexity. Monographs in Computer Science. Springer, New York (1999)
Dubois, O., Dequen, G.: A backbone-search heuristic for efficient solving of hard 3-SAT formulae. In: Nebel, B. (ed.) Proceedings of the Seventeenth International Joint Conference on Artificial Intelligence, IJCAI 2001, Seattle, Washington, USA, August 4-10, pp. 248–253 (2001)
Fellows, M.R., Hermelin, D., Rosamond, F.A., Vialette, S.: On the parameterized complexity of multiple-interval graph problems. Theoretical Computer Science 410(1), 53–61 (2009)
Fellows, M.R., Szeider, S., Wrightson, G.: On finding short resolution refutations and small unsatisfiable subsets. Theoretical Computer Science 351(3), 351–359 (2006)
Flum, J., Grohe, M.: Parameterized Complexity Theory. Texts in Theoretical Computer Science. An EATCS Series, vol. XIV. Springer, Berlin (2006)
Gallo, G., Longo, G., Pallotino, S., Nguyen, S.: Directed hypergraphs and applications. Discrete Applied Mathematics 42, 177–201 (1993)
Gwynne, M., Kullmann, O.: Generalising and unifying SLUR and unit-refutation completeness. In: van Emde Boas, P., Groen, F.C.A., Italiano, G.F., Nawrocki, J., Sack, H. (eds.) SOFSEM 2013. LNCS, vol. 7741, pp. 220–232. Springer, Heidelberg (2013)
Hertli, T., Moser, R.A., Scheder, D.: Improving PPSZ for 3-SAT using critical variables. In: Schwentick, T., Dürr, C. (eds.) Symposium on Theoretical Aspects of Computer Science, vol. 9, pp. 237–248. Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2011)
Hoos, H.H., Stützle, T.: SATLIB: An online resource for research on SAT. In: Gent, I., van Maaren, H., Walsh, T. (eds.) SAT 2000: Highlights of Satisfiability Research in the year 2000. Frontiers in Artificial Intelligence and Applications, pp. 283–292. Kluwer Academic (2000)
Impagliazzo, R., Paturi, R., Zane, F.: Which problems have strongly exponential complexity? J. of Computer and System Sciences 63(4), 512–530 (2001)
Kautz, H., Selman, B.: Pushing the envelope: planning, propositional logic, and stochastic search. In: Proceedings of the Thirteenth AAAI Conference on Artificial Intelligence, AAAI 1996, pp. 1194–1201. AAAI Press (1996)
Kilby, P., Slaney, J.K., Thiébaux, S., Walsh, T.: Backbones and backdoors in satisfiability. In: Proceedings of the Twentieth National Conference on Artificial Intelligence and the Seventeenth Innovative Applications of Artificial Intelligence Conference, AAAI 2005, Pittsburgh, Pennsylvania, USA, July 9-13, pp. 1368–1373 (2005)
Kullmann, O.: Investigating a general hierarchy of polynomially decidable classes of cnf’s based on short tree-like resolution proofs. Electronic Colloquium on Computational Complexity (ECCC) 6(41) (1999)
Kullmann, O.: An application of matroid theory to the SAT problem. In: Fifteenth Annual IEEE Conference on Computational Complexity, pp. 116–124 (2000)
Niedermeier, R.: Invitation to Fixed-Parameter Algorithms. Oxford Lecture Series in Mathematics and its. Applications. Oxford University Press, Oxford (2006)
Parkes, A.J.: Clustering at the phase transition. In: Proceedings of the Fourteenth National Conference on Artificial Intelligence, AAAI 1997, pp. 340–345. AAAI Press (1997)
Prelotani, D.: Efficiency and stability of hypergraph SAT algorithms. In: Johnson, D.S., Trick, M.A. (eds.) Cliques, Coloring and Satisfiability, pp. 479–498. AMS (1996)
Schneider, J., Froschhammer, C., Morgenstern, I., Husslein, T., Singer, J.M.: Searching for backbones – an efficient parallel algorithm for the traveling salesman problem. Computer Physics Communications 96, 173–188 (1996)
Slaney, J.K., Walsh, T.: Backbones in optimization and approximation. In: Nebel, B. (ed.) Proceedings of the Seventeenth International Joint Conference on Artificial Intelligence, IJCAI 2001, Seattle, Washington, USA, August 4-10, pp. 254–259 (2001)
Strichman, O.: Tuning SAT checkers for bounded model checking. In: Emerson, E.A., Sistla, A.P. (eds.) CAV 2000. LNCS, vol. 1855, pp. 480–494. Springer, Heidelberg (2000)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
de Haan, R., Kanj, I., Szeider, S. (2013). Local Backbones. In: Järvisalo, M., Van Gelder, A. (eds) Theory and Applications of Satisfiability Testing – SAT 2013. SAT 2013. Lecture Notes in Computer Science, vol 7962. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-39071-5_28
Download citation
DOI: https://doi.org/10.1007/978-3-642-39071-5_28
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-39070-8
Online ISBN: 978-3-642-39071-5
eBook Packages: Computer ScienceComputer Science (R0)