|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
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
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.|
|ps.gz-source:||dm060216.ps.gz (59 K)|
|ps-source:||dm060216.ps (160 K)|
|pdf-source:||dm060216.pdf (109 K)|
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