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Sampling from combinatorial families can be difficult. However, complicatedfamilies can often be embedded within larger, simpler ones, for which easysampling algorithms are known. We take advantage of such a relationship todescribe a sampling algorithm for the smaller family, via a Markov chainstarted at a random sample of the larger family. The utility of the method isdemonstrated via several examples, with particular emphasis on samplinglabelled graphs with given degree sequence, a well-studied problem for whichexisting algorithms leave much room for improvement. For graphs with givendegrees, with maximum degree $O(m^{1/4})$ where $m$ is the number of edges, weobtain an asymptotically uniform sample in $O(m)$ steps, which substantiallyimproves upon existing algorithms.

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