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Enumerating Contingency Tables via Random Permanents

Published online by Cambridge University Press:  01 January 2008

ALEXANDER BARVINOK
ALEXANDER BARVINOK*
Affiliation:
Department of Mathematics, University of Michigan, Ann Arbor, MI 48109-1043, USA (e-mail: barvinok@umich.edu)
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Abstract

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Given m positive integers R = (ri), n positive integers C = (cj) such that Σri = Σcj = N, and mn non-negative weights W=(wij), we consider the total weight T=T(R, C; W) of non-negative integer matrices D=(dij) with the row sums ri, column sums cj, and the weight of D equal to $\prod w_{ij}^{d_{ij}}$. For different choices of R, C, and W, the quantity T(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 in N and which computes a number T′=T′(R,C;W) such that T′ ≤ 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, ln T′ provides an asymptotically accurate estimate of ln T. The idea of the algorithm is to express T as the expectation of the permanent of an N × N random matrix with exponentially distributed entries and approximate the expectation by the integral T′ of an efficiently computable log-concave function on ℝmn.

Information

Type
Paper
Information
Combinatorics, Probability and Computing , Volume 17 , Issue 1 , January 2008 , pp. 1 - 19
Copyright
Copyright © Cambridge University Press 2007

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