Abstract
We consider boolean formulas in conjunctive normal form (CNF). If all clauses are large, it needs many clauses to obtain an unsatisfiable formula; moreover, these clauses have to interleave. We review quantitative results for the amount of interleaving required, many of which rely on the Lovász Local Lemma, a probabilistic lemma with many applications in combinatorics.
In positive terms, we are interested in simple combinatorial conditions which guarantee for a CNF formula to be satisfiable. The criteria obtained are nontrivial in the sense that even though they are easy to check, it is by far not obvious how to compute a satisfying assignment efficiently in case the conditions are fulfilled; until recently, it was not known how to do so. It is also remarkable that while deciding satisfiability is trivial for formulas that satisfy the conditions, a slightest relaxation of the conditions leads us into the territory of NP-completeness.
Several open problems remain, some of which we mention in the concluding section.
Research is supported by SNF Grant 200021-118001/1.
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.H.: The Probabilistic Method, 3rd edn. John Wiley & Sons Inc., Chichester (2008)
Erdős, P., Lovász, L.: Problems and results on 3-chromatic hypergraphs and some related questions. In: Hajnal, A., Rado, R., Sós, V.T. (eds.) Infinite and Finite Sets (to Paul Erdős on his 60th birthday), vol. II, pp. 609–627. North-Holland, Amsterdam (1975)
Erdős, P.: On a combinatorial problem. Nordisk Mat. Tidskr. 11, 5–10, 40 (1963)
Erdős, P.: On a combinatorial problem. II. Acta Math. Acad. Sci. Hungar. 15, 445–447 (1964)
Shearer, J.B.: On a problem of Spencer. Combinatorica 5(3), 241–245 (1985)
Beck, J.: An algorithmic approach to the Lovász Local Lemma. I. Random Struct. Algorithms 2(4), 343–365 (1991)
Alon, N.: A parallel algorithmic version of the local lemma. Random Struct. Algorithms 2(4), 367–378 (1991)
Czumaj, A., Scheideler, C.: A new algorithm approach to the general Lovász Local Lemma with applications to scheduling and satisfiability problems. In: Proc. 32nd Ann. ACM Symp. on Theory of Computing, pp. 38–47 (2000)
Srinivasan, A.: Improved algorithmic versions of the Lovász Local Lemma. In: Proc. 19th Ann. ACM-SIAM Symp. on Discrete Algorithms, pp. 611–620 (2008)
Moser, R.A.: Derandomizing the Lovász Local Lemma more effectively. CoRR abs/0807.2120 (2008)
Moser, R.A.: A constructive proof of the Lovász Local Lemma. CoRR abs/0810.4812 (2008); Proc. 41st Ann. ACM Symp. on Theory of Computing (to appear)
Moser, R.A., Tardos, G.: A constructive proof of the general Lovász Local Lemma. CoRR abs/0903.0544 (2009)
Erdős, P., Spencer, J.: Lopsided Lovász Local Lemma and Latin transversals. Discrete Appl. Math. 30(2-3), 151–154 (1991)
Lu, L., Székely, L.: Using Lovász Local Lemma in the space of random injections. Electron. J. Combin. 14(1), 13, Research Paper 63 (2007) (electronic)
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)
Tovey, C.A.: A simplified NP-complete satisfiability problem. Discrete Appl. Math. 8(1), 85–89 (1984)
Kratochvíl, J., Savický, P., Tuza, Z.: One more occurrence of variables makes satisfiability jump from trivial to NP-complete. SIAM J. Comput. 22(1), 203–210 (1993)
Savický, P., Sgall, J.: DNF tautologies with a limited number of occurrences of every variable. Theoret. Comput. Sci. 238(1-2), 495–498 (2000)
Hoory, S., Szeider, S.: A note on unsatisfiable k-CNF formulas with few occurrences per variable. SIAM J. Discrete Math. 20(2), 523–528 (2006)
Gebauer, H.: Disproof of the neighborhood conjecture with implications to SAT. CoRR abs/0904.2541 (2009)
Beck, J.: Combinatorial Games: Tic Tac Toe Theory, 1st edn. Encyclopedia of Mathematics and Its Applications, vol. 114. Cambridge University Press, Cambridge (2008)
Stříbrná, J.: Between Combinatorics and Formal Logic, Master’s Thesis. Charles University, Prague (1994)
Hoory, S., Szeider, S.: Computing unsatisfiable k-SAT instances with few occurences per variable. Theoret. Comput. Sci. 337(1-3), 347–359 (2005)
Kullmann, O.: New methods for 3-SAT decision and worst-case analysis. Theor. Comput. Sci. 223(1-2), 1–72 (1999)
Aharoni, R., Linial, N.: Minimal non-two-colorable hypergraphs and minimal unsatisfiable formulas. J. Comb. Theory, Ser. A 43(2), 196–204 (1986)
Davydov, G., Davydova, I., Kleine Büning, H.: An efficient algorithm for the minimal unsatisfiability problem for a subclass of CNF. Ann. Math. Artificial Intelligence 23(3-4), 229–245 (1998)
Kleine Büning, H.: An upper bound for minimal resolution refutations. In: Gottlob, G., Grandjean, E., Seyr, K. (eds.) CSL 1998. LNCS, vol. 1584, pp. 171–178. Springer, Heidelberg (1999)
Kleine Büning, H.: On subclasses of minimal unsatisfiable formulas. Discrete Appl. Math. 107(1-3), 83–98 (2000)
Szeider, S.: Homomorphisms of conjunctive normal forms. Discrete Appl. Math. 130(2), 351–365 (2003)
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)
Abbott, H.: An application of Ramsey’s Theorem to a problem of Erdős and Hajnal. Canad. Math. Bull. 8, 515–517 (1965)
Kostochka, A., Mubayi, D., Rödl, V., Tetali, P.: On the chromatic number of set systems. Random Struct. Algorithms 19(2), 87–98 (2001)
Scheder, D.: Unsatisfiable linear CNF formulas are large, and difficult to construct explicitely. CoRR abs/0905.1587 (2009)
Porschen, S., Speckenmeyer, E., Zhao, X.: Linear CNF formulas and satisfiability. Discrete Appl. Math. 157(5), 1046–1068 (2009)
Scheder, D.: Unsatisfiable linear k-CNFs exist, for every k. CoRR abs/0708.2336 (2007)
Ben-Sasson, E., Wigderson, A.: Short proofs are narrow—resolution made simple. J. ACM 48(2), 149–169 (2001)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2009 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Gebauer, H., Moser, R.A., Scheder, D., Welzl, E. (2009). The Lovász Local Lemma and Satisfiability. In: Albers, S., Alt, H., Näher, S. (eds) Efficient Algorithms. Lecture Notes in Computer Science, vol 5760. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-03456-5_3
Download citation
DOI: https://doi.org/10.1007/978-3-642-03456-5_3
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-03455-8
Online ISBN: 978-3-642-03456-5
eBook Packages: Computer ScienceComputer Science (R0)