The asymptotic volume of the Birkhoff polytope

TL;DR

This paper derives an asymptotic formula for the volume of the Birkhoff polytope and related transportation polytopes using recent enumeration results, as the matrix size grows large.

Contribution

It provides the first asymptotic volume formula for the Birkhoff polytope, extending to general transportation polytopes with variable dimensions.

Findings

Asymptotic volume formula for B(n) as n grows large

Extension of enumeration techniques to transportation polytopes

Insights into the geometric structure of high-dimensional polytopes

Abstract

Let m,n be positive integers. Define T(m,n) to be the transportation polytope consisting of the m x n non-negative real matrices whose rows each sum to 1 and whose columns each sum to m/n. The special case B(n)=T(n,n) is the much-studied Birkhoff-von Neumann polytope of doubly-stochastic matrices. Using a recent asymptotic enumeration of non-negative integer matrices (Canfield and McKay, 2007), we determine the asymptotic volume of T(m,n) as n goes to infinity, with m=m(n) such that m/n neither decreases nor increases too quickly. In particular, we give an asymptotic formula for the volume of B(n).