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Journal of Combinatorial Optimization

, Volume 22, Issue 3, pp 457–468 | Cite as

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

  • Ivona Bezáková
  • Nayantara Bhatnagar
  • Dana Randall
Article

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 2×2 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.

Keywords

Contingency tables Cell bounds Markov chains Mixing rates Fpras 

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Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  • Ivona Bezáková
    • 1
  • Nayantara Bhatnagar
    • 2
  • Dana Randall
    • 3
  1. 1.Department of Computer ScienceRochester Institute of TechnologyRochesterUSA
  2. 2.Department of StatisticsUniversity of CaliforniaBerkeleyUSA
  3. 3.College of ComputingGeorgia Institute of TechnologyAtlantaUSA

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