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Theoretical Computer Science 265 (2001) 159185 www.elsevier.com/locate/tcs

Summary: Theoretical Computer Science 265 (2001) 159­185
Lower bounds for random 3-SAT via di erential equations
Dimitris Achlioptas
Microsoft Research, Microsoft Corporation, One Microsoft Way, Redmond, WA 98052, USA
It is widely believed that the probability of satis˙ability for random k-SAT formulae exhibits
a sharp threshold as a function of their clauses-to-variables ratio. For the most studied case,
k = 3, there have been a number of results during the last decade providing upper and lower
bounds for the threshold's potential location. All lower bounds in this vein have been algorithmic,
i.e., in each case a particular algorithm was shown to satisfy random instances of 3-SAT with
probability 1 - o(1) if the clauses-to-variables ratio is below a certain value. We show how
di erential equations can serve as a generic tool for analyzing such algorithms by rederiving
most of the known lower bounds for random 3-SAT in a simple, uniform manner. c 2001
Elsevier Science B.V. All rights reserved.
Keywords: Random 3-sat; Algorithms; Di erential equations
1. Introduction
It is widely believed that the probability of satis˙ability for random k-SAT formu-
lae exhibits a sharp threshold as the ratio of clauses to variables is increased. More
precisely, let Fk(n; m) denote a random formula in Conjunctive Normal Form with


Source: Achlioptas, Dimitris - Department of Computer Engineering, University of California at Santa Cruz


Collections: Computer Technologies and Information Sciences