The "Burnside process" converges slowly
UNSPECIFIED (1998) The "Burnside process" converges slowly. In: 2nd International Workshop on Randomization and Approximation Techniques in Computer Science (RANDOM 98), BARCELONA, SPAIN, OCT 08-10, 1998. Published in: RANDOMIZATION AND APPROXIMATION TECHNIQUES IN COMPUTER SCIENCE, 1518 pp. 331-345.Full text not available from this repository.
We consider the problem of sampling "unlabelled structures", i.e., sampling combinatorial structures module 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 non-trivial 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 Ne 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.
|Item Type:||Conference Item (UNSPECIFIED)|
|Subjects:||Q Science > QA Mathematics > QA76 Electronic computers. Computer science. Computer software|
|Series Name:||LECTURE NOTES IN COMPUTER SCIENCE|
|Journal or Publication Title:||RANDOMIZATION AND APPROXIMATION TECHNIQUES IN COMPUTER SCIENCE|
|Editor:||Luby, M and Rolim, J and Serna, M|
|Number of Pages:||15|
|Page Range:||pp. 331-345|
|Title of Event:||2nd International Workshop on Randomization and Approximation Techniques in Computer Science (RANDOM 98)|
|Location of Event:||BARCELONA, SPAIN|
|Date(s) of Event:||OCT 08-10, 1998|
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