Partitions of the polytope of Doubly Substochastic Matrices

TL;DR

This paper explores three methods to partition the polytope of doubly substochastic matrices based on row/column sums, total sum, and sub-defect, and characterizes their extreme points and relations.

Contribution

It introduces three novel partitioning schemes of the doubly substochastic matrix polytope and characterizes the extreme points for each, enhancing understanding of its structure.

Findings

Three partitioning methods of the polytope are proposed.

Extreme points of each subpolytope are characterized.

Relations among extreme points in different partitions are established.

Abstract

In this paper, we provide three different ways to partition the polytope of doubly substochastic matrices into subpolytopes via the prescribed row and column sums, the sum of all elements and the sub-defect respectively. Then we characterize the extreme points of each type of convex subpolytopes. The relations of the extreme points of the subpolytopes in the three partitions are also given.