Generating random unlabelled graphs
SIAM Journal on Computing
Automating Po´lya theory: the computational complexity of the cycle index polynomial
Information and Computation
Surveys in combinatorics, 1995
The Markov chain Monte Carlo method: an approach to approximate counting and integration
Approximation algorithms for NP-hard problems
The Swendsen-Wang process does not always mix rapidly
STOC '97 Proceedings of the twenty-ninth annual ACM symposium on Theory of computing
Boltzmann Samplers, Pólya Theory, and Cycle Pointing
SIAM Journal on Computing
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We consider the problem of sampling ‘unlabelled structures’, i.e., sampling combinatorial structures modulo a group of symmetries. The main tool which has been used for this sampling problem is Burnside’s lemma. In situations where a significant proportion of the structures have no nontrivial symmetries, it is already fairly well understood how to apply this tool. More generally, it is possible to obtain nearly uniform samples by simulating a Markov chain that we call the Burnside process: this is a random walk on a bipartite graph which essentially implements Burnside’s lemma. For this approach to be feasible, the Markov chain ought to be ‘rapidly mixing’, i.e., converge rapidly to equilibrium. The Burnside process was known to be rapidly mixing for some special groups, and it has even been implemented in some computational group theory algorithms. In this paper, we show that the Burnside process is not rapidly mixing in general. In particular, we construct an infinite family of permutation groups for which we show that the mixing time is exponential in the degree of the group.