Fast and perfect sampling of subgraphs and polymer systems

We give an efficient perfect sampling algorithm for weighted, connected induced subgraphs (or graphlets) of rooted, bounded degree graphs under a vertex-percolation subcriticality condition. We show that this subcriticality condition is optimal in the sense that the problem of (approximately) sampling weighted rooted graphlets becomes impossible for infinite graphs and intractable for finite graphs if the condition does not hold. We apply our rooted graphlet sampling algorithm as a subroutine to give a fast perfect sampling algorithm for polymer models and a fast perfect sampling algorithm for weighted non-rooted graphlets in finite graphs, two widely studied yet very different problems. We apply this polymer model algorithm to give improved sampling algorithms for spin systems at low temperatures on expander graphs and other structured families of graphs: under the least restrictive conditions known we give near linear-time algorithms, while previous algorithms in these regimes required large polynomial running times.