Random Constraint Satisfaction Problems

Random instances of constraint satisfaction problems such as k-SAT provide challenging benchmarks. If there are m constraints over n variables there is typically a large range of densities r=m/n where solutions are known to exist with probability close to one due to non-constructive arguments. However, no algorithms are known to find solutions efficiently with a non-vanishing probability at even much lower densities. This fact appears to be related to a phase transition in the set of all solutions. The goal of this extended abstract is to provide a perspective on this phenomenon, and on the computational challenge that it poses

The Chromatic Number of Random Regular Graphs

Given any integer d >= 3, let k be the smallest integer such that d < 2k log k. We prove that with high probability the chromatic number of a random d-regular graph is k, k+1, or k+2, and that if (2k-1) \log k < d < 2k \log k then the chromatic number is either k+1 or k+2

Between 2- and 3-colorability

We consider the question of the existence of homomorphisms between $G_{n,p}$ and odd cycles when $p=c/n,\,1<c\leq 4$. We show that for any positive integer $\ell$, there exists $\epsilon=\epsilon(\ell)$ such that if $c=1+\epsilon$ then w.h.p. $G_{n,p}$ has a homomorphism from $G_{n,p}$ to $C_{2\ell+1}$ so long as its odd-girth is at least $2\ell+1$. On the other hand, we show that if $c=4$ then w.h.p. there is no homomorphism from $G_{n,p}$ to $C_5$. Note that in our range of interest, $\chi(G_{n,p})=3$ w.h.p., implying that there is a homomorphism from $G_{n,p}$ to $C_3$