Abstract
Let k be an even integer. We investigate the applicability of approximation techniques to the problem of deciding whether a random k-SAT formula is satisfiable. Let n be the number of propositional variables under consideration. First we show that if the number m of clauses satisfies m≥ Cn k/2 for a certain constant C, then unsatisfiability can be certified efficiently using (known) approximation algorithms for MAX CUT or MIN BISECTION. In addition, we present an algorithm based on the Lovász ϑ function that within polynomial expected time decides whether the input formula is satisfiable, provided m≥ Cn k/2. These results improve previous work by Goerdt and Krivelevich [14]. Finally, we present an algorithm that approximates random MAX 2-SAT within expected polynomial time.
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
Alon, N., Spencer, J.: The Probabilistic Method. John Wiley and Sons, Chichester (1992)
Ben-Sasson, E., Bilu, Y.: A Gap in Average Proof Complexity. In: ECCC 2003 (2002)
Chung, F.R.K.: Spectral Graph Theory. American Mathematical Society, Providence (1997)
Coja-Oghlan, A.: The Lovasz number of random graphs. Hamburger Beiträge zur Mathematik 169
Coja-Oghlan, A., Taraz, A.: Colouring random graphs in expected polynomial time. In: Alt, H., Habib, M. (eds.) STACS 2003. LNCS, vol. 2607, pp. 487–498. Springer, Heidelberg (2003)
Feige, U.: Relations between average case complexity and approximation complexity. In: Proc. 34th STOC, pp. 310–332 (2002)
Feige, U., Goemans, M.X.: Approximating the value of two prover proof systems, with applications to MAX 2SAT and MAX DICUT. In: Proc. 3rd Israel Symp. on Theory of Computing and Systems, pp. 182–189 (1995)
Feige, U., Krauthgamer, R.: A polylogarithmic approximation of the minimum bisection. In: Proc. 41st FOCS, pp. 105–115 (2000)
Feige, U., Ofek, E.: Spectral techniques applied to sparse random graphs, report MCS03-01, Weizmann Institute of Science (2003)
Friedgut, E.: Necessary and Sufficient Conditions for Sharp Thresholds of Graph Properties and the k-SAT problem. J. Amer. Math. Soc. 12, 1017–1054 (1999)
Friedman, J., Goerdt, A.: Recognizing more unsatisfiable random 3-SAT instances efficiently. In: Orejas, F., Spirakis, P.G., van Leeuwen, J. (eds.) ICALP 2001. LNCS, vol. 2076, p. 310. Springer, Heidelberg (2001)
Goemans, M.X., Williamson, D.P.: Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming. J. ACM 42, 1115–1145
Goerdt, A., Jurdzinski, T.: Some results on random unsatisfiable k-sat instances and approximation algorithms applied to random structures. In: Diks, K., Rytter, W. (eds.) MFCS 2002. LNCS, vol. 2420, pp. 280–291. Springer, Heidelberg (2002)
Goerdt, A., Krivelevich, M.: Efficient recognition of random unsatisfiable k-SAT instances by spectral methods. In: Ferreira, A., Reichel, H. (eds.) STACS 2001. LNCS, vol. 2010, pp. 294–304. Springer, Heidelberg (2001)
Grötschel, M., Lovász, L., Schrijver, A.: Geometric algorithms and combinatorial optimization. Springer, Heidelberg (1988)
Hofri, M.: Probabilistic Analysis of Algorithms. Springer, Heidelberg (1987)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2003 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Coja-Oghlan, A., Goerdt, A., Lanka, A., Schädlich, F. (2003). Certifying Unsatisfiability of Random 2k-SAT Formulas Using Approximation Techniques. In: Lingas, A., Nilsson, B.J. (eds) Fundamentals of Computation Theory. FCT 2003. Lecture Notes in Computer Science, vol 2751. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-45077-1_3
Download citation
DOI: https://doi.org/10.1007/978-3-540-45077-1_3
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-40543-6
Online ISBN: 978-3-540-45077-1
eBook Packages: Springer Book Archive