Fast and perfect sampling of subgraphs and polymer systems

TL;DR

This paper introduces an efficient perfect sampling algorithm for weighted connected subgraphs and polymer systems, leveraging a vertex-percolation process, with applications to spin systems and graphlet sampling.

Contribution

It presents a novel perfect sampling method for subgraphs and polymer models that is optimal under certain conditions and improves existing algorithms for specific graph classes.

Findings

Sampling is impossible in finite expected time without the condition.

The algorithm achieves near linear-time sampling for polymer models.

Improves sampling for spin systems on specific graph classes.

Abstract

We give an efficient perfect sampling algorithm for weighted, connected induced subgraphs (or graphlets) of rooted, bounded degree graphs. Our algorithm utilizes a vertex-percolation process with a carefully chosen rejection filter and works under a percolation subcriticality condition. We show that this condition is optimal in the sense that the task of (approximately) sampling weighted rooted graphlets becomes impossible in finite expected time for infinite graphs and intractable for finite graphs when the condition does not hold. We apply our sampling algorithm as a subroutine to give near linear-time perfect sampling algorithms for polymer models and weighted non-rooted graphlets in finite graphs, two widely studied yet very different problems. This new perfect sampling algorithm for polymer models gives improved sampling algorithms for spin systems at low temperatures on expander graphs and unbalanced bipartite graphs, among other applications.