 
Summary: On the Maximum Satisfiability of Random Formulas
Dimitris Achlioptas
Department of Computer Science, University of California, Santa Cruz
optas@cs.ucsc.edu
Assaf Naor
Microsoft Research, Redmond, Washington
anaor@microsoft.com
Yuval Peres
Microsoft Research, Redmond, Washington
peres@stat.berkeley.edu
Abstract
Say that a kCNF a formula is psatisfiable if there exists a truth assignment satisfying a fraction
1  2k
+ p 2k
of its clauses (note that every kCNF formula is 0satisfiable). Let Fk(n, m) denote a
random kCNF formula on n variables with m clauses. For every k 2 and every r > 0 we determine p
and = (k) = O(k2k/2
) such that with probability tending to 1 as n , a random kCNF formula
Fk(n, rn) is psatisfiable but not (p + )satisfiable.
1 Introduction
