A random polynomial-time algorithm for approximating the volume of convex bodies
Journal of the ACM (JACM)
The Markov chain Monte Carlo method: an approach to approximate counting and integration
Approximation algorithms for NP-hard problems
On sampling with Markov chains
Proceedings of the seventh international conference on Random structures and algorithms
Random Structures & Algorithms
Random walks and an O*(n5) volume algorithm for convex bodies
Random Structures & Algorithms
Random generation of 2 × n contingency tables
Random Structures & Algorithms
Polynomial-time counting and sampling of two-rowed contingency tables
Theoretical Computer Science
Improved Bounds for Sampling Contingency Tables
RANDOM-APPROX '99 Proceedings of the Third International Workshop on Approximation Algorithms for Combinatorial Optimization Problems: Randomization, Approximation, and Combinatorial Algorithms and Techniques
Approximate counting by dynamic programming
Proceedings of the thirty-fifth annual ACM symposium on Theory of computing
Proceedings of the twenty-fourth ACM SIGMOD-SIGACT-SIGART symposium on Principles of database systems
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We consider the problem of counting the number of contingency tables with given row and column sums. This problem is known to be #P-complete, even when there are only two rows [7]. In this paper we present the first fully-polynomial randomized approximation scheme for counting contingency tables when the number of rows is constant. A novel feature of our algorithm is that it is a hybrid of an exact counting technique with an approximation algorithm, giving two distinct phases. In the first, the columns are partitioned into "small" and "large". We show that the number of contingency tables can be expressed as the weighted sum of a polynomial number of new instances of the problem, where each instance consists of some new row sums and the original large column sums. In the second phase, we show how to approximately count contingency tables when all the column sums are large. In this case, we show that the solution lies in approximating the volume of a single convex body, a problem which is known to be solvable in polynomial time [5].