Random generation of combinatorial structures from a uniform
Theoretical Computer Science
Approximate counting, uniform generation and rapidly mixing Markov chains
Information and Computation
A random polynomial-time algorithm for approximating the volume of convex bodies
Journal of the ACM (JACM)
A polynomial time algorithm for counting integral points in polyhedra when the dimension is fixed
Mathematics of Operations Research
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
On Barvinok's algorithm for counting lattice points in fixed dimension
Mathematics of Operations Research
Random generation of 2 × n contingency tables
Random Structures & Algorithms
Polynomial-time counting and sampling of two-rowed contingency tables
Theoretical Computer Science
Rapidly Mixing Markov Chains for Sampling Contingency Tables with a Constant Number of Rows
FOCS '02 Proceedings of the 43rd Symposium on Foundations of 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
Approximately counting integral flows and cell-bounded contingency tables
Proceedings of the thirty-seventh annual ACM symposium on Theory of computing
Enumerating contingency tables via random permanents
Combinatorics, Probability and Computing
On the Diaconis-Gangolli Markov Chain for Sampling Contingency Tables with Cell-Bounded Entries
COCOON '09 Proceedings of the 15th Annual International Conference on Computing and Combinatorics
An invariance principle for polytopes
Proceedings of the forty-second ACM symposium on Theory of computing
Approximately Counting Integral Flows and Cell-Bounded Contingency Tables
SIAM Journal on Computing
On the Diaconis-Gangolli Markov chain for sampling contingency tables with cell-bounded entries
Journal of Combinatorial Optimization
An invariance principle for polytopes
Journal of the ACM (JACM)
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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 (Random Structures Algorithms 10(4) (1997) 487). 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 (J. ACM 38 (1) (1991) 1).