Discrete Mathematics & Theoretical Computer Science, Vol 6, No 2 (2004)

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DMTCS vol 6 no 2 (2004), pp. 425-436

Discrete Mathematics & Theoretical Computer Science


Volume 6 n° 2 (2004), pp. 425-436

author:Puyhaubert, Vincent
title:Generating functions and the satisfiability threshold
keywords:First moment method, exponential generating functions, saddle-point bounds
abstract:The 3-SAT problem consists in determining if a boolean formula with 3 literals per clause is satisfiable. When the ratio between the number of clauses and the number of variables increases, a threshold phenomenon is observed: the probability of satisfiability appears to decrease sharply from 1 to 0 in the neighbourghood of a threshold value, conjectured to be close to 4.25. Although the threshold has been proved to exist for the 2-SAT formulæ and for closely related problems like 3-XORSAT, there is still no proof for the 3-SAT problem. Recent works have provided so far upper and lower bounds for the threshold's potential location. We present here a unified approach to upper bounds that is based on urn models, generating functions, and saddle-point bounds. In this way, we re-derive some of the most significant upper bounds known in a simple and uniform manner.

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reference: Puyhaubert, Vincent (2004), Generating functions and the satisfiability threshold, Discrete Mathematics and Theoretical Computer Science 6, pp. 425-436
bibtex:For a corresponding BibTeX entry, please consider our BibTeX-file.
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Automatically produced on Thu Oct 14 21:52:57 CEST 2004 by falk