Abstract. Given m positive integers R = (ri), n positive integers C = (cj) such that ∑ ri = ∑ cj = N, and mn non-negative weights W = (wij), we consider the total weight T(R, C; W) of non-negative integer matrices (contingency tables) D = (dij) with the row sums ri, column sums cj, and the weight of D equal to ∏ w dij ij. We present a randomized algorithm of a polynomial in N complexity which approximates T(R, C; W) within a factor of (2πN) −1/2 (2πt) N/2teN/12t2 where t = max{minri, mincj}. In many cases, this approximation provides an asymptotically accurate estimate of ln T(R, C; W). The idea of the algorithm is to express T(R, C; W) as the expectation of the permanent of an N × N random matrix with exponentially distributed entries and approximate the expectation by the integral of an efficiently computable log-concave function on Rmn. 1. Introduction and