Enumeration and constructions of vertices of the polytope of polystochastic matrices
Anna A. Taranenko
TL;DR
This work advances the understanding of the Birkhoff polytope $\Omega_n^d$ for small dimensions by fully enumerating vertices of $\Omega_3^4$ (24 vertices across 21 equivalence classes with a notable symmetric subset) and $\Omega_4^3$ (533 equivalence classes, including symmetric cases), correcting prior results. It introduces algorithmic strategies that link vertex structure to supports via incidence matrices, and demonstrates new vertex-construction techniques based on multidimensional matrix products, including a broad family of symmetric vertices for $\Omega_3^d$ with explicit size formulas for the support and detailed permanence properties. The paper also connects these combinatorial objects to hypergraph fractional transversals, providing a unified framework for generating and analyzing vertices through Kronecker and dot products and hyperplane-structured constructions. Overall, the results yield both exact enumerations for small cases and scalable construction methods that push toward understanding vertices of high-dimensional polystochastic polytopes and their permanents.
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 $Ω_n^d$ known as the Birkhoff polytope. In this paper, we identify all vertices of the polytopes $Ω_4^3$ and $Ω_3^4$ correcting the results of Ke, Li, and Xiao (2016). Additionally, we describe constructions vertices of $Ω_n^d$ using multidimensional matrix products and find symmetric vertices of $Ω_3^d$ for all $d \geq 4$ with large support sizes.