Enumerating Contingency Tables via Random Permanents

Abstract

Givenmpositive integersR= (ri),npositive integersC= (cj) such that Σri= Σcj=N, andmnnon-negative weightsW=(wij), we consider the total weightT=T(R, C;W) of non-negative integer matricesD=(dij) with the row sumsri, column sumscj, and the weight ofDequal to$\prod w_{ij}^{d_{ij}}$. For different choices ofR,C, andW, the quantityT(R,C;W) specializes to the permanent of a matrix, the number of contingency tables with prescribed margins, and the number of integer feasible flows in a network. We present a randomized algorithm whose complexity is polynomial inNand which computes a numberT′=T′(R,C;W) such thatT′ ≤T≤ α(R,C)T′ where$\alpha(R,C) = \min \bigl\{\prod r_i! r_i^{-r_i}, \ \prod c_j! c_j^{-c_j} \bigr\} N^N/N!$. In many cases, lnT′ provides an asymptotically accurate estimate of lnT. The idea of the algorithm is to expressTas the expectation of the permanent of anN×Nrandom matrix with exponentially distributed entries and approximate the expectation by the integralT′ of an efficiently computable log-concave function on ℝmn.