More Vertices of the Tristochastic Polytope

TL;DR

This paper investigates the vertex structure of the tristochastic polytope, revealing it has exponentially many vertices, vastly exceeding the number of Latin squares, thus extending classical results about doubly stochastic matrices.

Contribution

It establishes that the tristochastic polytope has at least exponential in the number of Latin squares many vertices, showing a richer structure than previously understood.

Findings

The tristochastic polytope has at least $L_n^{2-o(1)}$ vertices.

Latin squares form a small subset of the polytope's vertices.

The vertex count grows exponentially with n.

Abstract

The $n\times n$ doubly stochastic matrices constitute a polytope in $\mathbb{R}^{n^2}$, and by Birkhoff's theorem, its vertex set coincides with the set of order-$n$ permutation matrices.\\ A tristochastic array is an $n \times n\times n$ array of nonnegative reals, where each row, column, and shaft sums to one. These arrays constitute a polytope $\Delta_n$ in $\mathbb{R}^{n^3}$. In analogy, it is easy to see that each of the $L_n$ order-$n$ Latin squares is a vertex of $\Delta_n$, but in contrast to Birkhoff's theorem, Latin squares form a vanishingly small subset of $\Delta_n$'s vertex set. We show here that $\Delta_n$ has at least $L_n^{2-o(1)}$ vertices.