Support of extremal doubly stochastic arrays

TL;DR

This work completely determines the minimal support size of $n\times m$ doubly stochastic arrays as $n+m-\gcd(n,m)$ and shows that, when $\gcd(n,m)=1$, all extremal arrays have the minimal possible support $n+m-1$. For the case $m=kn+1$, it provides a complete, checkable structure theorem: a subset $\Omega$ is the support of an extremal array iff each row has exactly $k+1$ nonzeros and $\Omega$ contains no cycles, with a constructive proof via a tree-based weight assignment. The paper also characterizes extremal arrays with non-minimal support sizes (showing all sizes $n+m-s$ with $1\le s\le \gcd(n,m)$ occur) and, in the $m=n+1$ case, yields a precise enumeration of extremal arrays and a correspondence with labeled trees, leading to the existence of non-equivalent extremals sharing the same multiset of entries for $n\ge 6$. These results bridge combinatorial graph methods, matrix extremality, and tiling interpretations in abelian groups, with implications for joint distributions and transportation-type problems.

Abstract

An $n \times m$ array with nonnegative entries is called doubly stochastic if the sum of its entries at each row is $m$ and at each column is $n$. The set of all $n \times m$ doubly stochastic arrays is a convex polytope with finitely many extremal points. The main result of this paper characterizes the possible sizes of the supports of all extremal $n \times m$ doubly stochastic arrays. In particular we prove that the minimal size of the support of an $n \times m$ doubly stochastic array is $n + m - \gcd(n,m)$. Moreover, for $m=kn+1$ we also characterize the structure of the support of the extremal arrays.

Figures (6)

• Figure 2.1: The graphs associated to the two matrices in \ref{['eqDS1.3']}.
• Figure 3.1: The discrete trapezoid function for $n=5$, $m=7$.
• Figure 4.1: Graph prototypes of type I (left) and type II (right).
• Figure 4.2: Graph prototypes of type III (left) and type IV (right).
• Figure 6.1: An example of two trees on $7$ vertices which correspond to two non-equivalent extremal $6 \times 7$ arrays with the same multiset of entries.

Theorems & Definitions (31)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6
• Proposition 2.1: see Li96
• Proposition 2.2: see Li96
• Proposition 2.3: see Car96
• Proposition 2.4