SAT 2008: Theory and Applications of Satisfiability Testing – SAT 2008 pp 246-256 | Cite as
How Many Conflicts Does It Need to Be Unsatisfiable?
Abstract
A pair of clauses in a CNF formula constitutes a conflict if there is a variable that occurs positively in one clause and negatively in the other. Clearly, a CNF formula has to have conflicts in order to be unsatisfiable—in fact, there have to be many conflicts, and it is the goal of this paper to quantify how many.
An unsatisfiable k-CNF has at least 2 k clauses; a lower bound of 2 k for the number of conflicts follows easily. We improve on this trivial bound by showing that an unsatisfiable k-CNF formula requires Ω(2.32 k ) conflicts. On the other hand there exist unsatisfiable k-CNF formulas with \(O(\frac{4^k\log^3 k}{k})\) conflicts. This improves the simple bound O(4 k ) arising from the unsatisfiable k-CNF formula with the minimum number of clauses.
Keywords
satisfiability unsatisfiable formulas conflict graph Lovász Local LemmaPreview
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References
- 1.Kullmann, O.: The combinatorics of conflicts between clauses. In: Giunchiglia, E., Tacchella, A. (eds.) SAT 2003. LNCS, vol. 2919, pp. 426–440. Springer, Heidelberg (2004)Google Scholar
- 2.Erdős, P., Faudree, R.J., Rousseau, C.C., Schelp, R.H.: The size ramsey number. Periodica Mathematica Hungarica 9(2-2), 145–161 (1978)CrossRefMathSciNetGoogle Scholar
- 3.Faudree, R.J., Schelp, R.H.: A survey of results on the size Ramsey number. In: Paul Erdős and his mathematics, II (Budapest, 1999), János Bolyai Math. Soc. Budapest. Bolyai Soc. Math. Stud, vol. 11, pp. 291–309 (2002)Google Scholar
- 4.Hoory, S., Szeider, S.: A note on unsatisfiable k-CNF formulas with few occurrences per variable. SIAM Journal on Discrete Mathematics 20(2), 523–528 (2006)CrossRefMathSciNetMATHGoogle Scholar
- 5.Welzl, E.: Boolean satisfiability – combinatorics and algorithms (lecture notes) (2005), http://www.inf.ethz.ch/~emo/SmallPieces/SAT.ps
- 6.Kratochvíl, J., Savický, P., Tuza, Z.: One more occurrence of variables makes satisfiability jump from trivial to NP-complete. SIAM Journal of Computing 22(1), 203–210 (1993)CrossRefMATHGoogle Scholar
- 7.Berman, P., Karpinski, M., Scott, A.D.: Approximation hardness and satisfiability of bounded occurrence instances of SAT. Electronic Colloquium on Computational Complexity (ECCC) 10(022) (2003)Google Scholar
- 8.Hooker, J.N., Vinay, V.: Branching rules for satisfiability. Journal of Automated Reasoning 15, 359–383 (1995)CrossRefMathSciNetMATHGoogle Scholar
- 9.Czumaj, A., Scheideler, C.: A new algorithm approach to the general Lovász local lemma with applications to scheduling and satisfiability problems (extended abstract). In: Proceedings of the Thirty-Second Annual ACM Symposium on Theory of Computing, pp. 38–47. ACM, New York (2000); (electronic) CrossRefGoogle Scholar
- 10.Erdős, P., Spencer, J.: Lopsided Lovász local lemma and Latin transversals. Discrete Appl. Math. 30, 151–154 (1991); ARIDAM III (New Brunswick, NJ, 1988) CrossRefMathSciNetGoogle Scholar
- 11.Alon, N., Spencer, J.H.: The probabilistic method, 2nd edn. Wiley-Interscience Series in Discrete Mathematics and Optimization. Wiley-Interscience [John Wiley & Sons], New York (2000); With an appendix on the life and work of Paul ErdősMATHGoogle Scholar
- 12.Lu, L., Székely, L.: Using Lovász local lemma in the space of random injections. Electron. J. Combin. 14(1), 13 (2007) Research Paper 63 (electronic) Google Scholar
- 13.Papadimitriou, C.H., Wolfe, D.: The complexity of facets resolved. J. Comput. Syst. Sci. 37(1), 2–13 (1988)CrossRefMathSciNetMATHGoogle Scholar
- 14.Szeider, S.: Minimal unsatisfiable formulas with bounded clause-variable difference are fixed-parameter tractable. J. Comput. Syst. Sci. 69(4), 656–674 (2004)CrossRefMathSciNetMATHGoogle Scholar
- 15.Kleine Büning, H.: On subclasses of minimal unsatisfiable formulas. Discrete Appl. Math. 107(1-3), 83–98 (2000); Boolean functions and related problemsCrossRefMathSciNetMATHGoogle Scholar
- 16.Fleischner, H., Kullmann, O., Szeider, S.: Polynomial-time recognition of minimal unsatisfiable formulas with fixed clause-variable difference. Theoret. Comput. Sci. 289(1), 503–516 (2002)CrossRefMathSciNetMATHGoogle Scholar
- 17.Kleine Büning, H., Zhao, X.: The complexity of some subclasses of minimal unsatisfiable formulas. J. Satisf. Boolean Model. Comput. 3(1-2), 1–17 (2007)MathSciNetMATHGoogle Scholar