On the hardness of sampling independent sets beyond the tree threshold

Abstract

We consider local Markov chain Monte-Carlo algorithms for sampling from the weighted distribution of independent sets with activity $ł$, where the weight of an independent set $I$ is $ł^{|I|}$. A recent result has established that Gibbs sampling is rapidly mixing in sampling the distribution for graphs of maximum degree $d$ and $ł<ł_c(d)$, where $ł_c(d)$ is the critical activity for uniqueness of the Gibbs measure (i.e., for decay of correlations with distance in the weighted distribution over independent sets) on the $d$-regular infinite tree. We show that for $d \geq 3$, $ł$ just above $ł_c(d)$ with high probability over $d$-regular bipartite graphs, any local Markov chain Monte-Carlo algorithm takes exponential time before getting close to the stationary distribution. Our results provide a rigorous justification for ``replica'' method heuristics. These heuristics were invented in theoretical physics and are used in order to derive predictions on Gibbs measures on random graphs in terms of Gibbs measures on trees. We conjecture that $ł_c$ is in fact the exact threshold for this computational problem, i.e., that for $ł>ł_c$ it is NP-hard to approximate the above weighted sum overindependent sets to within a factor polynomial in the size of the graph.