# 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