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arXiv · 0705.2422

The asymptotic volume of the Birkhoff polytope

E. Rodney Canfield·Brendan D. McKay
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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).

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BibTeXRIS

E. Rodney Canfield, Brendan D. McKay. 2007-05-16. The asymptotic volume of the Birkhoff polytope. https://arxiv.org/abs/0705.2422

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