Permanents of doubly substochastic matrices

Cao, Lihong; Zhi, Chen; Koyuncu, Selcuk; Li, Huilan

doi:10.1080/03081087.2018.1513448

Cited by 6 publications

(6 citation statements)

References 11 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…, 1) ∈ R n , and 0 ≤ σ ≤ n, ∆(R,S, σ) is the set of all doubly substochastic matrices with total sum equal to σ. By Lemma 4.2, for all A ∈ ∆(R,S, σ), per(A) ≤ 1, where per(A) = 1 if and only if σ = n and A is a permutation matrix ( [10,21]).…”

Section: (S) =ρ(E(s))mentioning

confidence: 98%

Extreme Points of Certain Transportation Polytopes with Fixed Total Sums

Chen

Zhu

et al. 2021

ELA

Self Cite

View full text Add to dashboard Cite

Transportation matrices are $m\times n$ nonnegative matrices with given row sum vector $R$ and column sum vector $S$. All such matrices form the convex polytope $\mathcal{U}(R,S)$ which is called a transportation polytope and its extreme points have been classified. In this article, we consider a new class of convex polytopes $\Delta(\bar{R},\bar{S},\sigma)$ consisting of certain transportation polytopes satisfying that the sum of all elements is $\sigma$, and the row and column sum vectors are dominated componentwise by the given positive vectors $\bar{R}$ and $\bar{S}$, respectively. We characterize the extreme points of $\Delta(\bar{R},\bar{S},\sigma)$. Moreover, we give the minimal term rank and maximal permanent of $\Delta(\bar{R},\bar{S},\sigma)$.

show abstract

Section: (S) =ρ(E(s))mentioning

confidence: 98%

Extreme Points of Certain Transportation Polytopes with Fixed Total Sums

Chen

Zhu

et al. 2021

ELA

Self Cite

View full text Add to dashboard Cite

show abstract

“…Denote by per(A) the permanent of A. The following results were given in [10] and [8] respectively. 1, [8]).…”

Section: Some Applicationsmentioning

confidence: 99%

“…Denote by per(A) the permanent of A. The following results were given in [10] and [8] respectively. Theorem 7.3 can also be rephrased with respect to the sub-defect k as the following corollary.…”

Section: Some Applicationsmentioning

confidence: 99%

Partitions of the polytope of doubly substochastic matrices

Cao

Zhi

2019

Linear Algebra and its Applications

Self Cite

View full text Add to dashboard Cite

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 ω n . The set Ω n is a convex polytope, and has been intensively studied by many mathematicians [11,12,13,15,16,17,18,23,24,25,33,34,32]. Specially, the extreme points of Ω n are exactly the permutation matrices due to Birkhoff [4] and von Neumann [31], which can be stated as follows.Theorem 1.3. [4, 31] An n × n matrix A is a doubly stochastic matrix if and only if there are finite permutation matrices P 1 , P 2 , · · · , P N and positive numbers α 1 , · · · , α N such that α 1 + · · · + α N = 1 and A = α 1 P 1 + · · · + α N P N .The set ω n is also a convex polytope and its extreme points are partial permutation matrices [21], i.e., matrices with at most one element in each row and each column equal to one and other elements zero. Since Ω n ⊆ ω n , one may wonder if the classical 2010 Mathematics Subject Classification. 15B51; 52B05; 05A18.

show abstract

“…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%