 
Summary: Why a Random 3SAT Problem is Harder Than Another?
Chu Min Li & Anbulagan
LaRIA, Universite de Picardie Jules Verne, 33, Rue St. Leu, 80039 Amiens Cedex 01, France
tel: (33) 03 22 82 78 75 fax: (33) 03 22 82 75 02 email: fcli@laria.upicardie.fr Anbulagan@utc.frg
Abstract: We study the mean height and width of search trees constructed by a highly opti
mized DPL procedure to solve hard random 3SAT problems. Based on experimentation we found
that the evolution of DPL search trees follows some simple laws and a problem is harder because
there is no or very few short sequences of assumptions (partial assignments under construction of
truth values to variables) allowing to falsify it so that the corresponding search tree is wider.
1 Introduction
Consider a propositional formula F in Conjunctive Normal Form (CNF) on a set of Boolean vari
ables fx1 x2 ::: xng, the satis ability (SAT) problem consists in testing whether clauses in F can
all be satis ed by some consistent assignment of truth values (1 or 0) to the variables. If it is
the case, F is said satis able otherwise, F is said unsatis able. If each clause exactly contains r
literals, the subproblem is called rSAT problem. 3SAT is the smallest NPcomplete subproblem
of SAT. SAT problem is fundamental in many elds of computer science, electrical engineering and
mathematics.
It is well known that random 3SAT problems are very hard to solve on the average when the
ratio of clausetovariable is approximatively equal to 4.25 7, 1] and some problems are much harder
than others. The phenomenon appears to be beyond argument when one uses an algorithm based
