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arXiv · 2502.09149

Enumeration and constructions of vertices of the polytope of polystochastic matrices

Anna A. Taranenko
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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.

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Anna A. Taranenko. 2025-02-13. Enumeration and constructions of vertices of the polytope of polystochastic matrices. https://arxiv.org/abs/2502.09149

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