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Tail Bounds for Occupancy and the Satisfiability Threshold Conjecture
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BibTeX
@MISC{Kamath_tailbounds,
author = {Anil Kamath and Rajeev Motwani and et al.},
title = {Tail Bounds for Occupancy and the Satisfiability Threshold Conjecture},
year = {}
}
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Abstract
The classical occupancy problem is concerned with studying the number of empty bins resulting from a random allocation of m balls to n bins. We provide a series of tail bounds on the distribution of the number of empty bins. These tail bounds should find application in randomized algorithms and probabilistic analysis. Our motivating application is the following well-known conjecture on thresh-old phenomenon for the satisfiability problem. Consider random 3-SAT formulas with cn clauses over n variables, where each clause is chosen uniformly and independently from the space of all clauses of size 3. It has been conjectured that there is a sharp threshold for satisfiability at c * x 4.2. We provide the first non-trivial upper bound on the value of c*, showing that for c> 4.758 a random 3-SAT formula is unsatisfiable with high probability. This result is based on a structural property, possibly of independent interest, whose proof needs several applications of the occupancy tail bounds.
Keyphrases
tail bound satisfiability threshold conjecture random 3-sat formula empty bin thresh-old phenomenon randomized algorithm several application classical occupancy problem cn clause ran-dom allocation occupancy tail bound following well-known conjecture satisfiability problem inde-pendent interest first non-trivial upper bound structural property high probability sharp threshold probabilistic analysis
