An approximation algorithm for counting contingency tables

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 well as between the largest and the smallest column sums are strictly smaller than the golden ratio (1 + $ {sqrt{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. © 2010 Wiley Periodicals, Inc. Random Struct. Alg., 2010