 
Summary: Theoretical Computer Science 265 (2001) 159185
www.elsevier.com/locate/tcs
Lower bounds for random 3SAT via di erential equations
Dimitris Achlioptas
Microsoft Research, Microsoft Corporation, One Microsoft Way, Redmond, WA 98052, USA
Abstract
It is widely believed that the probability of satis˙ability for random kSAT formulae exhibits
a sharp threshold as a function of their clausestovariables 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 3SAT with
probability 1  o(1) if the clausestovariables 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 3SAT in a simple, uniform manner. c 2001
Elsevier Science B.V. All rights reserved.
Keywords: Random 3sat; Algorithms; Di erential equations
1. Introduction
It is widely believed that the probability of satis˙ability for random kSAT 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
