On the Vertices of the d-Dimensional Birkhoff Polytope

Abstract

Let us denote by Ωn the Birkhoff polytope of n×n doubly stochastic matrices. As the Birkhoff---von Neumann theorem famously states, the vertex set of Ωn coincides with the set of all n×n permutation matrices. Here we consider a higher-dimensional analog of this basic fact. Let $\varOmega^{(2)}_{n}$ be the polytope which consists of all tristochastic arrays of order n. These are n×n×n arrays with nonnegative entries in which every line sums to 1. What can be said about $\varOmega ^{(2)}_{n}$'s vertex set? It is well known that an order-n Latin square may be viewed as a tristochastic array where every line contains nź1 zeros and a single 1 entry. Indeed, every Latin square of order n is a vertex of $\varOmega^{(2)}_{n}$, but as we show, such vertices constitute only a vanishingly small subset of $\varOmega^{(2)}_{n}$'s vertex set. More concretely, we show that the number of vertices of $\varOmega ^{(2)}_{n}$ is at least $(L_{n})^{\frac{3}{2}-o(1)}$, where Ln is the number of order-n Latin squares. We also briefly consider similar problems concerning the polytope of n×n×n arrays where the entries in every coordinate hyperplane sum to 1, improving a result from Kravtsov (Cybern. Syst. Anal., 43(1):25---33, 2007). Several open questions are presented as well.