Rapidly Mixing Markov Chains for Sampling Contingency Tables with a Constant Number of Rows

We consider the problem of sampling almost uniformly from the set of contingency tables with given row and column sums, when the number of rows is a constant. Cryan and Dyer [3] have recently given a fully polynomial randomized approximation scheme (fpras) for the related counting problem, which only employs Markov chain methods indirectly. But they leave open the question as to whether a natural Markov chain on such tables mixes rapidly. Here we answer this question in the affirmative, and hence provide a very different proof of the main result of [3]. We show that the “¤¦¥§¤ heat-bath” Markov chain is rapidly mixing. We prove this by considering first a heat-bath chain operating on a larger window. Using techniques developed by Morris and Sinclair [20] (see also Morris [19]) for the multidimensional knapsack problem, we show that this chain mixes rapidly. We then apply the comparison method of Diaconis and Saloff-Coste [8] to show that the ¤¨¥¦¤ chain is rapidly mixi...

• Type: Conference
• Year: 2002
• Where: FOCS
• Authors: Mary Cryan, Martin E. Dyer, Leslie Ann Goldberg, Mark Jerrum, Russell A. Martin
• Keywords: FOCS 2002, Markov Chain, Markov Chain Methods, Natural Markov Chain, Theoretical Computer Science