 
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
Amin CojaOghlan
School of Informatics, University of Edinburgh
acoghlan@inf.ed.ac.uk
Federico RicciTersenghi
Physics Department, University of Rome "La Sapienza"
federico.ricci@roma1.infn.it
Abstract
For various random constraint satisfaction problems there is a significant gap between the largest constraint den
sity for which solutions exist and the largest density for which any polynomial time algorithm is known to find
solutions. Examples of this phenomenon include random kSAT, random graph coloring, and a number of other
random Constraint Satisfaction Problems. To understand this gap, we study the structure of the solution space of
random kSAT (i.e., the set of all satisfying assignments viewed as a subgraph of the Hamming cube). We prove
that for densities well below the satisfiability threshold, the solution space decomposes into an exponential number
of connected components and give quantitative bounds for the diameter, volume and number.
1 Introduction
