Solution clustering in random satisfiability
Department of Computer Science, University of California Santa Cruz, 1156 High Street MS:SOE3, 95064 Santa Cruz, USA
Corresponding author: a firstname.lastname@example.org
Revised: 22 May 2008
Published online: 15 August 2008
For a large number of random constraint satisfaction problems, such as random k-SAT and random graph and hypergraph coloring, we have very good estimates of the largest constraint density for which solutions exist. All known polynomial-time algorithms for these problems, though, 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 in random k-SAT for , 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 – Spin-glass and other random models
© EDP Sciences, Società Italiana di Fisica, Springer-Verlag, 2008