Enumeration and constructions of vertices of the polytope of polystochastic matrices

TL;DR

This paper characterizes all vertices of specific high-dimensional polytopes of polystochastic matrices, correcting previous results and introducing new construction methods for vertices, including symmetric ones with large support.

Contribution

It identifies all vertices of the polytopes _4^3 and _3^4, corrects earlier findings, and develops new constructions for vertices using multidimensional matrix products.

Findings

Vertices of _4^3 and _3^4 are fully characterized.

New methods for constructing vertices of _n^d are proposed.

Symmetric vertices of _3^d with large support are identified.

Abstract

A multidimensional nonnegative matrix is called polystochastic if the sum of entries in each of its lines equals $1$. The set of all polystochastic matrices of order $n$ and dimension $d$ is a convex polytope $\Omega_n^d$ known as the Birkhoff polytope. In this paper, we identify all vertices of the polytopes $\Omega_4^3$ and $\Omega_3^4$ correcting the results of Ke, Li, and Xiao (2016). Additionally, we describe constructions vertices of $\Omega_n^d$ using multidimensional matrix products and find symmetric vertices of $\Omega_3^d$ for all $d \geq 4$ with large support sizes.