Vincent Puyhaubert - Generating functions and the satisfiability threshold

dmtcs:325 - Discrete Mathematics & Theoretical Computer Science, January 1, 2004, Vol. 6 no. 2 - https://doi.org/10.46298/dmtcs.325
Generating functions and the satisfiability threshold

Authors: Vincent Puyhaubert

    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.


    Volume: Vol. 6 no. 2
    Published on: January 1, 2004
    Imported on: March 26, 2015
    Keywords: First moment method,exponential generating functions,saddle-point bounds,[INFO.INFO-DM] Computer Science [cs]/Discrete Mathematics [cs.DM]

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