Abstract
We address the problem of enumerating all models of Boolean formulæ in order of non-decreasing weight in Schaefer’s framework. The weight of a model is the number of variables assigned to 1. Tractability in this context amounts to enumerating all models one after the other in sorted order, with polynomial delay between two successive outputs. The question of model-enumeration has already been studied in Schaefer’s framework, but without imposing a specific order. The order of non-decreasing weight changes the complexity considerably. We obtain a new dichotomous complexity classification. On the one hand, we develop new polynomial delay algorithms for Horn and 2-XOR-formulæ to enumerate the models by non-decreasing weight. On the other hand, we prove that in all other cases such a polynomial delay algorithm does not exist, unless P=NP.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Bagan, G., Durand, A., Grandjean, E.: On acyclic conjunctive queries and constant delay enumeration. In: Duparc, J., Henzinger, T.A. (eds.) CSL 2007. LNCS, vol. 4646, pp. 208–222. Springer, Heidelberg (2007)
Barg, A.: Complexity issues in coding theory. Electronic Colloquium on Computational Complexity (ECCC) 4(46) (1997)
Cohen, D.A.: Tractable decision for a constraint language implies tractable search. Constraints 9(3), 219–229 (2004)
Creignou, N., Hébrard, J.-J.: On generating all solutions of generalized satisfiability problems. Theoretical Informatics and Applications 31(6), 499–511 (1997)
Creignou, N., Kolaitis, P.G., Vollmer, H. (eds.): Complexity of Constraints. LNCS, vol. 5250. Springer, Heidelberg (2008)
Creignou, N., Schnoor, H., Schnoor, I.: Nonuniform boolean constraint satisfaction problems with cardinality constraint. ACM Trans. Comput. Log. 11(4) (2010)
Creignou, N., Vollmer, H.: Boolean constraint satisfaction problems: When does Post’s lattice help? In: Creignou, et al [5], pp. 3–37
Hagen, M.: Lower bounds for three algorithms for transversal hypergraph generation. Discrete Applied Mathematics (2009)
Jeavons, P.G.: On the algebraic structure of combinatorial problems. Theoretical Computer Science 200, 185–204 (1998)
Johnson, D.S., Papadimitriou, C.H., Yannakakis, M.: On generating all maximal independent sets. Inf. Process. Lett. 27(3), 119–123 (1988)
Jonsson, P., Nordh, G.: Introduction to the maximum solution problem. In: Creignou, et al [5], pp. 255–282
Khanna, S., Sudan, M., Williamson, D.: A complete classification of the approximability of maximization problems derived from Boolean constraint satisfaction. In: Proceedings 29th Symposium on Theory of Computing, pp. 11–20. ACM Press, New York (1997)
Khuller, S., Vazirani, V.V.: Planar graph coloring is not self-reducible, assuming P≠NP. Theoretical Computer Science 88(1), 183–189 (1991)
Kimelfeld, B., Sagiv, Y.: Incrementally computing ordered answers of acyclic conjunctive queries. In: Etzion, O., Kuflik, T., Motro, A. (eds.) NGITS 2006. LNCS, vol. 4032, pp. 141–152. Springer, Heidelberg (2006)
Kimelfeld, B., Sagiv, Y.: Efficiently enumerating results of keyword search over data graphs. Inf. Syst. 33(4-5), 335–359 (2008)
Krokhin, A.A., Marx, D.: On the hardness of losing weight. In: Aceto, L., Damgård, I., Goldberg, L.A., Halldórsson, M.M., Ingólfsdóttir, A., Walukiewicz, I. (eds.) ICALP 2008, Part I. LNCS, vol. 5125, pp. 662–673. Springer, Heidelberg (2008)
Makino, K., Uno, T.: New algorithms for enumerating all maximal cliques. In: Hagerup, T., Katajainen, J. (eds.) SWAT 2004. LNCS, vol. 3111, pp. 260–272. Springer, Heidelberg (2004)
Marx, D.: Parameterized complexity of constraint satisfaction problems. Computational Complexity 14(2), 153–183 (2005)
Schaefer, T.J.: The complexity of satisfiability problems. In: Proccedings 10th Symposium on Theory of Computing, pp. 216–226. ACM Press, New York (1978)
Schmidt, J.: Enumeration: Algorithms and complexity. Preprint (2009), http://www.thi.uni-hannover.de/fileadmin/forschung/arbeiten/schmidt-da.pdf
Schnoor, H., Schnoor, I.: Enumerating all solutions for constraint satisfaction problems. In: Thomas, W., Weil, P. (eds.) STACS 2007. LNCS, vol. 4393, pp. 694–705. Springer, Heidelberg (2007)
Schnoor, H., Schnoor, I.: Partial polymorphisms and constraint satisfaction problems. In: Creignou, et al [5], pp. 229–254
Schnorr, C.P.: Optimal algorithms for self-reducible problems. In: International Conference on Automata, Languages and Programming, pp. 322–337 (1976)
Strozecki, Y.: Enumeration complexity and matroid decomposition. Phd thesis (2010)
Vazirani, V., Yannakakis, M.: Suboptimal cuts: Their enumeration, weight and number. In: Kuich, W. (ed.) ICALP 1992. LNCS, vol. 623, pp. 366–377. Springer, Heidelberg (1992)
Yeh, L.-P., Wang, B.-F., Su, H.-H.: Efficient algorithms for the problems of enumerating cuts by non-decreasing weights. Algorithmica 56(3), 297–312 (2010)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2011 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Creignou, N., Olive, F., Schmidt, J. (2011). Enumerating All Solutions of a Boolean CSP by Non-decreasing Weight. In: Sakallah, K.A., Simon, L. (eds) Theory and Applications of Satisfiability Testing - SAT 2011. SAT 2011. Lecture Notes in Computer Science, vol 6695. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-21581-0_11
Download citation
DOI: https://doi.org/10.1007/978-3-642-21581-0_11
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-21580-3
Online ISBN: 978-3-642-21581-0
eBook Packages: Computer ScienceComputer Science (R0)