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

• Authors: Mary Cryan;Martin E. Dyer;Leslie Ann Goldberg;Mark Jerrum;Russell A. Martin
• Venue: FOCS '02 Proceedings of the 43rd Symposium on Foundations of Computer Science
• Year: 2002

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 "2 脳 2 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 2 脳 2 chain is rapidly mixing. As part of our analysis, we give the first proof that the 2 脳 2 chain mixes in time polynomial in the input size when both the number of rows and the number of columns is constant.