Let us denote by Ω n the Birkhoff polytope of n×n doubly stochastic matrices. As the Birkhoff-von Neumann theorem famously states, the vertex set of Ω n coincides with the set of all n×n permutation matrices. Here we consider a higher-dimensional analog of this basic fact. Let Ωn(2) 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 Ωn(2)'s vertex set? It is well known that an order-n Latin square may be viewed as a tristochastic array where every line contains n-1 zeros and a single 1 entry. Indeed, every Latin square of order n is a vertex of Ωn(2), but as we show, such vertices constitute only a vanishingly small subset of Ωn(2)'s vertex set. More concretely, we show that the number of vertices of Ωn(2) is at least (Ln)3/2-o(1), where L n 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.
Bibliographical noteFunding Information:
N. Linial was supported by ISF and BSF grants.
- Birkhoff polytope
- Latin squares