Abstract
Some basics of combinatorial block design are combined with certain constraint satisfaction problems of interest to the satisfiability community. The paper shows how such combinations lead to satisfiability problems, and shows empirically that these are some of the smallest very hard satisfiability problems ever constructed. Partially balanced (0,1) designs (PB01Ds) are introduced as an extension of balanced incomplete block designs (BIBDs) and (0,1) designs. Also, (0,1) difference sets are introduced as an extension of certain cyclical difference sets. Constructions based on (0,1) difference sets enable generation of PB01Ds over a much wider range of parameters than is possible for BIBDs. Building upon previous work of Spence, it is shown how PB01Ds lead to small, very hard, unsatisfiable formulas. A new three-dimensional form of combinatorial block design is introduced, and leads to small, very hard, satisfiable formulas. The methods are validated on solvers that performed well in the SAT 2009 and earlier competitions.
This is a preview of subscription content, log in via an institution to check access.
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
Anderson, I.: Combinatorial Designs: Construction Methods. Ellis Horwood (1990)
Colbourn, C.J., Dinitz, J.H. (eds.): CRC Handbook of Combinatorial Designs. CRC Press, Boca Raton (1996)
Eén, N., Sörensson, N.: MiniSat – A SAT Solver with Conflict-Clause Minimization. In: Bacchus, F., Walsh, T. (eds.) SAT 2005. LNCS, vol. 3569. Springer, Heidelberg (2005)
Gebser, M., Kaufmann, B., Neumann, A., Schaub, T.: Conflict-driven answer set solving. In: IJCAI, pp. 386–392. AAAI, Menlo Park (2007)
Kullmann, O.: Polynomial time SAT decision for complementation-invariant clause-sets, and sign-non-singular matrices. In: Marques-Silva, J., Sakallah, K.A. (eds.) SAT 2007. LNCS, vol. 4501, pp. 314–327. Springer, Heidelberg (2007)
Le Berre, D.: The SAT competitions (2009), http://www.satcompetition.org/
Porschen, S., Schmidt, T.: On some SAT-variants over linear formulas. In: Nielsen, M., Kucera, A., Miltersen, P.B., Palamidessi, C., Tuma, P., Valencia, F.D. (eds.) SOFSEM 2009. LNCS, vol. 5404, pp. 449–460. Springer, Heidelberg (2009)
Spence, I.: tts: A SAT-solver for small, difficult instances. Journal on Satisfiability, Boolean Modeling and Computation 4, 173–190 (2008)
Spence, I.: sgen1: A generator of small but difficult satisfiability benchmarks. Journal of Experimental Algorithms 15, 1.1–1.15 (2010)
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2010 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Van Gelder, A., Spence, I. (2010). Zero-One Designs Produce Small Hard SAT Instances. In: Strichman, O., Szeider, S. (eds) Theory and Applications of Satisfiability Testing – SAT 2010. SAT 2010. Lecture Notes in Computer Science, vol 6175. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-14186-7_37
Download citation
DOI: https://doi.org/10.1007/978-3-642-14186-7_37
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-14185-0
Online ISBN: 978-3-642-14186-7
eBook Packages: Computer ScienceComputer Science (R0)