 
Summary: On the SolutionSpace Geometry
of Random Constraint Satisfaction Problems
Dimitris Achlioptas
Department of Computer Science
University of California Santa Cruz
optas@cs.ucsc.edu
Federico RicciTersenghi
Department of Physics
University of Rome "La Sapienza"
federico.ricci@roma1.infn.it
ABSTRACT
For a number of random constraint satisfaction problems, such as
random kSAT and random graph/hypergraph coloring, there are
very good estimates of the largest constraint density for which so
lutions exist. Yet, all known polynomialtime algorithms for these
problems fail to find solutions even 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 prove that much before solutions disap
pear, they organize into an exponential number of clusters, each
