### Abstract

We present a randomized approximation algorithm for counting contingency tables, m × n non-negative integer matrices with given row sums R = (r_{1},...,r_{m})and column sums C = (c_{1},...,c_{n}). We define smooth margins (R,C) in terms of the typical table and prove that for such margins the algorithm has quasi-polynomial N^{O(ln N)} complexity, where N = r_{1} + + r_{m} = c_{1} + +c_{n}. Various classes of margins are smooth, e.g., when m = O(n), n = O(m) and the ratios between the largest and the smallest row sums as we√l as between the largest and the smallest column sums are strictly smaller than the golden ratio (1 + √ 5)/2 ≈ 1.618. The algorithm builds on Monte Carlo integration and sampling algorithms for log-concave densities, the matrix scaling algorithm, the permanent approximation algorithm, and an integral representation for the number of contingency tables.

Original language | English (US) |
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Pages (from-to) | 25-66 |

Number of pages | 42 |

Journal | Random Structures and Algorithms |

Volume | 37 |

Issue number | 1 |

DOIs | |

State | Published - Aug 1 2010 |

### Keywords

- Contingency tables
- Matrix scaling
- Permanent approximation
- Randomized algorithms

### ASJC Scopus subject areas

- Software
- Mathematics(all)
- Computer Graphics and Computer-Aided Design
- Applied Mathematics

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## Cite this

*Random Structures and Algorithms*,

*37*(1), 25-66. https://doi.org/10.1002/rsa.20301