Polynomial-time counting and sampling of two-rowed contingency tables

In this paper a Markov chain for contingency tables with two rows is defined. The chain is shown to be rapidly mixing using the path coupling method. We prove an upper bound for the mixing time of the chain. The upper bound is quadratic in the number of columns and linear in the logarithm of the table sum. By considering a specific problem, we prove a lower bound for the mixing time of the chain. The lower bound is quadratic in the number of columns and linear in the logarithm of the number of columns. A fully polynomial randomised approximation scheme for this problem is described.

[1]  Diane Hernek.  Random generation of 2 × n contingency tables , 1998 .

[2]  N. S. Barnett,et al.  Private communication , 1969 .

[3]  Mark Jerrum,et al.  The Markov chain Monte Carlo method: an approach to approximate counting and integration , 1996 .

[4]  Martin E. Dyer,et al.  Path coupling: A technique for proving rapid mixing in Markov chains , 1997, Proceedings 38th Annual Symposium on Foundations of Computer Science.

[5]  Martin E. Dyer,et al.  A more rapidly mixing Markov chain for graph colorings , 1998, Random Struct. Algorithms.

[6]  P. Diaconis,et al.  Testing for independence in a two-way table , 1985 .

[7]  Martin E. Dyer,et al.  A Genuinely Polynomial-Time Algorithms for Sampling Two-Rowed Contingency Tables , 1998, ICALP.

[8]  Prasad Tetali,et al.  Simple Markov-chain algorithms for generating bipartite graphs and tournaments , 1997, SODA '97.

[9]  Dorit S. Hochbaum,et al.  Approximation Algorithms for NP-Hard Problems , 1996 .

[10]  M. Dyer.  Computing the volume of convex bodies : a case where randomness provably helps , 1991 .

[11]  Rajeev Motwani,et al.  Randomized Algorithms , 1995, SIGA.

[12]  D. Aldous.  Random walks on finite groups and rapidly mixing markov chains , 1983 .

[13]  R. Graham,et al.  On sampling with Markov chains , 1996 .

[14]  Jim Albert,et al.  Estimation in Contingency Tables Using Prior Information , 1983 .

[15]  Martin E. Dyer,et al.  Sampling contingency tables , 1997 .

[16]  A. Scott,et al.  The Analysis of Categorical Data from Complex Sample Surveys: Chi-Squared Tests for Goodness of Fit and Independence in Two-Way Tables , 1981 .

[17]  Diane Hernek.  Random generation of 2×n contingency tables , 1998, Random Struct. Algorithms.

[18]  W. Schroeder,et al.  Graphs, Dynamic Programming, and Finite Games , 1968 .

[19]  Martin Dyer,et al.  A more rapidly mixing Markov chain for graph colorings , 1998 .

[20]  P. Diaconis,et al.  Rectangular Arrays with Fixed Margins , 1995 .

[21]  Richard M. Karp,et al.  Monte-Carlo algorithms for enumeration and reliability problems , 1983, 24th Annual Symposium on Foundations of Computer Science (sfcs 1983).

[22]  Martin E. Dyer,et al.  Beating the 2Δ bound for approximately counting colourings: a computer-assisted proof of rapid mixing , 1998, SODA '98.