Support of extremal doubly stochastic arrays

TL;DR

This paper characterizes the support sizes of extremal doubly stochastic arrays, establishing the minimal support size as n + m - gcd(n,m), and describes their structure for specific cases.

Contribution

It provides a complete characterization of the support sizes of extremal doubly stochastic arrays and details their structure in certain parameter regimes.

Findings

Minimal support size is n + m - gcd(n,m).

Supports of extremal arrays have specific structural properties when m=kn+1.

The set of extremal arrays forms a convex polytope with finitely many extremal points.

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.