Expand and Contract: Sampling graphs with given degrees and other combinatorial families

Abstract: Sampling from combinatorial families can be difficult. However, complicated families can often be embedded within larger, simpler ones, for which easy sampling algorithms are known. We take advantage of such a relationship to describe a sampling algorithm for the smaller family, via a Markov chain started at a random sample of the larger family. The utility of the method is demonstrated via several examples, with particular emphasis on sampling labelled graphs with given degree sequence, a well-studied problem for which existing algorithms leave much room for improvement. For graphs with given degrees, with maximum degree $O(m^{1/4})$ where $m$ is the number of edges, we obtain an asymptotically uniform sample in $O(m)$ steps, which substantially improves upon existing algorithms.