2019
DOI: 10.1016/j.laa.2018.10.024
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Partitions of the polytope of doubly substochastic matrices

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.Denote the set of all n × n doubly stochastic matrices by Ω n , and the set of all n × n doubly substochastic matrices by … Show more

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Cited by 9 publications
(3 citation statements)
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“…This representation relates to the problem of characterizing the extreme points of the polytope of doubly substochastic n × n matrices with entries summing to an specified integer. This polytope was studied in Cao and Chen (2019), where Theorem 4.1 shows that the extreme points of interest are exactly the partial permutation matrices of order n − k. This implies the existence of a coupling for W ε p (µ, ν) p whose marginals are uniform over n − k = (1 − ε)n points (giving mass 1/n to each point).…”
Section: A8 Proof Of Propositionmentioning
confidence: 99%
“…This representation relates to the problem of characterizing the extreme points of the polytope of doubly substochastic n × n matrices with entries summing to an specified integer. This polytope was studied in Cao and Chen (2019), where Theorem 4.1 shows that the extreme points of interest are exactly the partial permutation matrices of order n − k. This implies the existence of a coupling for W ε p (µ, ν) p whose marginals are uniform over n − k = (1 − ε)n points (giving mass 1/n to each point).…”
Section: A8 Proof Of Propositionmentioning
confidence: 99%
“…Moreover, they figured out its dual Hopf algebra (KS, x * G , µ, ∆ * , ν) and a closed-formula of its antipode [28]. The symmetric group on a set X, denoted as Sym(X), is a group of all bijections from X to itself, which is widely applied to many areas, such as algebraic number theory [29] and substochastic matrices [30][31][32][33]. In combinatorics, a permutation of degree n is an arrangement of n elements.…”
Section: Introductionmentioning
confidence: 99%
“…Later on, Chen, Cao and Wang [11] characterized the extreme points of the polytope of centrosymmetric doubly substochastic matrices. Moreover, Cao and Chen [9] studied the convex set ω s n of all n × n doubly substochastic matrices with the sum of all elements equal to s.…”
mentioning
confidence: 99%