Abstract
k-CNF is the class of CNF formulas in which the length of each clause of every formula is k. The decision problem asks for an assignment of truth values to the variables that satisfies all the clauses of a given CNF formula. k-SAT problem is k-CNF decision problem. Cook [9] has shown that k-SAT is NP-complete for k ≥ 3. LCNF is the class of linear formulas and LSAT is its decision problem. In [3] we present a general method to construct linear minimal unsatisfiable (MU) formulas. NP = PCP(log,1) is called PCP theorem, and it is equivalent to that there exists some r > 1 such that (3SAT, r-UN3SAT)(or denoted by \((1-\frac{1}{r})-GAP3SAT)\) is NP-complete [1,2]. In this paper,we show that for k ≥ 3, (kSAT, r-UNkSAT) is NP-completre and (LSAT, r-UNLSAT) is NP-completre for some r > 1. Based on the application of linear MU formulas, [3] we construct a reduction from (4SAT, r-UN4SAT) to (LSAT ≥ 4,r′-UNLSAT ≥ 4), and proved that (LSAT ≥ 4,r − UNLSAT ≥ 4) is NP-complete for some r > 1, so the approximation problem s-Approx-LSAT ≥ 4 is NP-hard for some s > 1.
The work is supported by the National Natural Science Foundation of China (No. 60563008) and the Special Foundation for Improving Science Research Condition of Guizhou Province of China.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Du, D., Ko, K.-I., Wang, J.: Introduction to Computational Complexity (Chinese). Higher Education Press, P.R.China (2004)
Arora, S., Barak, B.: Computational Complexity: A Modern Approach. Princeton University, Princeton (2007)
Zhang, Q., Xu, D.: The Existence of Unsatisfiable Formulas in k-LCNF for k ≥ 3. In: Cai, J.-Y., Cooper, S.B., Zhu, H. (eds.) TAMC 2007. LNCS, vol. 4484, pp. 616–623. Springer, Heidelberg (2007)
Davydov, G., Davydova, I., Kleine Büning, H.: An efficient algorithm for the minimal unsatisfiability problem for a subclass of CNF. Annals of Mathematics and Artificial Intelligence 23, 229–245 (1998)
Fleischner, H., Kullmann, O., Szeider, S.: Polynomial-time recognition of minimal unsatisfiable formulas with fixed clause-variable difference. Theoretical Computer Science 289(1), 503–516 (2002)
Porschen, S., Speckenmeyer, E.: Linear CNF formulas and satisfiability, Tech. Report zaik2006-520, University Köln (2006)
Porschen, S., Speckenmeyer, E., Randerath, B.: On Linear CNF Formulas. In: Biere, A., Gomes, C.P. (eds.) SAT 2006. LNCS, vol. 4121, pp. 212–225. Springer, Heidelberg (2006)
Porschen, S., Speckenmeyer, E.: NP-completeness of SAT for restricted linear formulas classes. In: Proceedings of Guangzhou Symposium on Satisfiability in Logic-Based Modeling, vol. 1, pp. 108–121, pp. 111–123 (1997)
Cook, S.C.: The complexity of theorem-proving procedures. In: Proc. 3rd ACM STOC, pp. 151–158 (1971)
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 2008 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Deng, T., Xu, D. (2008). NP-Completeness of (k-SAT,r-UNk-SAT) and (LSAT ≥ k ,r-UNLSAT ≥ k ). In: Preparata, F.P., Wu, X., Yin, J. (eds) Frontiers in Algorithmics. FAW 2008. Lecture Notes in Computer Science, vol 5059. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-69311-6_11
Download citation
DOI: https://doi.org/10.1007/978-3-540-69311-6_11
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-69310-9
Online ISBN: 978-3-540-69311-6
eBook Packages: Computer ScienceComputer Science (R0)