 
Summary: EPJ manuscript No.
(will be inserted by the editor)
Solution Clustering in Random Satisfiability
Dimitris Achlioptas a
Department of Computer Science, University of California Santa Cruz
Received: date / Revised version: date
Abstract. For a large number of random constraint satisfaction problems, such as random kSAT and
random graph and hypergraph coloring, we have very good estimates of the largest constraint density for
which solutions exist. All known polynomialtime algorithms for these problems, though, already fail to
find solutions at much lower densities. To understand the origin of this gap we study how the structure
of the space of solutions evolves in such problems as constraints are added. In particular, we show that
for k 8, much before solutions disappear, they organize into an exponential number of clusters, each of
which is relatively small and far apart from all other clusters. Moreover, inside each cluster most variables
are frozen, i.e., take only one value.
PACS. 02.50.r Probability theory, stochastic processes, and statistics 75.10.Nr Spinglass and other
random models
1 Introduction
For a number of random Constraint Satisfaction Prob
lems (CSP), we have very good rigorous estimates for
the largest constraint density (ratio of constraints to vari
