Asymptotic bounds on the numbers of vertices of polytopes of polystochastic matrices

TL;DR

The paper studies the vertex count V(n,d) of the high-dimensional Birkhoff polytope Ω_n^d of polystochastic matrices. It derives a general upper bound via polytope-face theory and reviews known asymptotics, showing a wide gap between lower and upper bounds for fixed n, especially at n=3. It then proves a new lower bound for V(3,d) that grows doubly exponentially in d by constructing a large family of polystochastic matrices using epsilon-sparse sets, zero-sum matrices, and trifferent codes, ensuring many distinct vertices arise from their decompositions. This demonstrates the rich combinatorial structure of Ω_n^d and highlights the sharp contrast between small fixed-order cases and high dimension.

Abstract

A multidimensional nonnegative matrix is called polystochastic if the sum of entries in each line is equal to $1$. The set of all polystochastic matrices of order $n$ and dimension $d$ is a convex polytope $Ω_n^d$. In the present paper, we compare known bounds on the number of vertices of the polytope $Ω_n^d$ and prove that the number of vertices of $Ω_3^d$ is doubly exponential on $d$.

Theorems & Definitions (19)

• Proposition 1: see, e.g., Bron.covexpoly
• Proposition 2
• proof
• Theorem 1
• Theorem 2: Keevash.existdesII, LinLur.hdimper
• Theorem 3: LinLur.birvert
• Proposition 3
• proof
• Conjecture 1
• Theorem 4: PotKrot.numbernary