 
Summary: Random Formulas Have Frozen Variables
Dimitris Achlioptas
Department of Computer Science, University of California Santa Cruz
optas@cs.ucsc.edu
Federico RicciTersenghi
Physics Department, University of Rome "La Sapienza"
federico.ricci@roma1.infn.it
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. Yet,
all known polynomialtime algorithms for these problems fail to find solutions even at much lower densities. To
understand the origin of this gap one can study how the structure of the space of solutions evolves in such problems as
constraints are added. In particular, it is known that 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. Here we further prove that
inside every cluster a majority of variables are frozen, i.e., take only one value. The existence of such frozen variables
gives a satisfying intuitive explanation for the failure of the polynomialtime algorithms analyzed so far. At the
same time, our results lend support to one of the two main hypotheses underlying Survey Propagation, a heuristic
introduced by physicists in recent years that appears to perform extraordinarily well on random constraint satisfaction
problems.
1 Introduction
