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 = (r1, . . . , rm) and column sums C = (c1, . . . , cn). We define smooth margins (R,C) in terms of the typical table and prove that for such margins the algorithm has quasi-polynomial NO(lnN) complexity, where N = r1 + · · · + rm = c1 + · · · + cn. 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+√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., 37, 25–66, 2010