Enumerating contingency tables via random permanents, preprint arXiv.pdfVIP

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Enumerating contingency tables via random permanents, preprint arXiv.pdf

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Enumerating contingency tables via random permanents, preprint arXiv

ENUMERATING CONTINGENCY TABLES VIA RANDOM PERMANENTS Alexander Barvinok March 2006 Abstract. Given m positive integers R = (ri), n positive integers C = (cj) such that P ri = P cj = N , and mn non-negative weights W = (wij), we consider the total weight T = T (R,C;W ) of non-negative integer matrices (contingency ta- bles) D = (dij) with the row sums ri, column sums cj , and the weight of D equal to Q w dij ij . We present a randomized algorithm of a polynomial in N complexity which computes a number T ′ = T ′(R,C;W ) such that T ′ ≤ T ≤ α(R,C)T ′ where α(R,C) = min nQ ri!r ?ri i , Q cj !c ?cj j o NN/N !. In many cases, lnT ′ provides an asymptotically accurate estimate of lnT . 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 effi- ciently computable log-concave function on Rmn. Applications to counting integer flows in graphs are also discussed. 1. Introduction and main results (1.1) Contingency tables. Let us fix m positive integers r1, . . . , rm and n posi- tive integers c1, . . . , cn such that r1 + . . .+ rm = c1 + . . .+ cn = N. A non-negative m × n integer matrix D = (dij) with the row sums r1, . . . , rm and the column sums c1, . . . , cn is called a contingency table with the margins R = (r1, . . . , rm) and C = (c1, . . . , cn). The problem of efficient enumeration of contingency tables with prescribed margins has attracted a lot of attention recently, see [DG95], [D+97], [CD03], [Mo02], [C+05]. The interest in contingency tables 1991 Mathematics Subject Classification. 05A16, 68R05, 60C05. Key words and phrases. contingency tables, permanent, randomized algorithms, log-concave functions. This research was partially supported by NSF Grant DMS 0400617. The author is grateful to Microsoft (Redmond) for hospitality during his work on this paper. Typeset by AMS-TEX 1 is motivated by applications to statistics,