On the Diaconis-Gangolli Markov chain for sampling contingency tables with cell-bounded entries

Authors:
Ivona Bezáková;Nayantara Bhatnagar;Dana Randall
Affiliations:
Department of Computer Science, Rochester Institute of Technology, Rochester, USA;Department of Statistics, University of California, Berkeley, USA;College of Computing, Georgia Institute of Technology, Atlanta, USA
Venue:
Journal of Combinatorial Optimization
Year:
2011

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Abstract

The problems of uniformly sampling and approximately counting contingency tables have been widely studied, but efficient solutions are only known in special cases. One appealing approach is the Diaconis and Gangolli Markov chain which updates the entries of a random 2脳2 submatrix. This chain is known to be rapidly mixing for cell-bounded tables only when the cell bounds are all 1 and the row and column sums are regular. We demonstrate that the chain can require exponential time to mix in the cell-bounded case, even if we restrict to instances for which the state space is connected. Moreover, we show the chain can be slowly mixing even if we restrict to natural classes of problem instances, including regular instances with cell bounds of either 0 or 1 everywhere, and dense instances where at least a linear number of cells in each row or column have non-zero cell-bounds.