Asymptotic estimates for the number of contingency tables, integer flows, and volumes of transportation polytopes

TL;DR

This paper develops efficient asymptotic estimates for counting contingency tables, integer flows, and computing volumes of transportation polytopes, revealing positive correlations under certain conditions.

Contribution

It introduces convex optimization-based asymptotic estimates for counting and volume calculations related to contingency tables and transportation polytopes, with new correlation insights.

Findings

Estimates are computationally efficient via convex optimization.

Asymptotic positive correlation between row and column sum events.

Results apply when row and column sums are far from uniform.

Abstract

We prove an asymptotic estimate for the number of mxn non-negative integer matrices (contingency tables) with prescribed row and column sums and, more generally, for the number of integer feasible flows in a network. Similarly, we estimate the volume of the polytope of mxn non-negative real matrices with prescribed row and column sums. Our estimates are solutions of convex optimization problems and hence can be computed efficiently. As a corollary, we show that if row sums R=(r_1, ..., r_m) and column sums C=(c_1, ..., c_n) with r_1 + ... + r_m =c_1 + ... +c_n =N are sufficiently far from constant vectors, then, asymptotically, in the uniform probability space of the mxn non-negative integer matrices with the total sum N of entries, the event consisting of the matrices with row sums R and the event consisting of the matrices with column sums C are positively correlated.