On Counting Independent Sets in Sparse Graphs

Abstract

We prove two results concerning approximate counting of independent sets in graphs with constant maximum degree $\Delta$. The first implies that the Markov chain Monte Carlo technique is likely to fail if $\Delta \geq 6$. The second shows that no fully polynomial randomized approximation scheme can exist for $\Delta \geq 25$, unless $\mathrm{RP}=\mathrm{NP}$.