Minimum permanents of doubly stochastic matrices with prescribed zero entries on the main diagonal
type: Item
instance of: scholarly article
title: Minimum permanents of doubly stochastic matrices with prescribed zero entries on the main diagonal
zbMATH Open document ID: 0807.15004
DOI: 10.1016/0024-3795(94)90111-2
published in: Linear Algebra and its Applications
publication date: 1994-08-15T00:00:00Z
review text: Let \(Z = \{(i_ 1, i_ 1), \dots, (i_ k, i_ k)\}\) be a subset of \(\{1, \dots, n\} \times \{1, \dots, n\}\); \(\Omega_ n(Z)\) denotes the doubly stochastic matrices with zeros at least in the positions of \(Z\). A matrix \(A \in \Omega_ n (Z)\) is said to be (permanent) minimizing in \(\Omega_ n (Z)\) if \(\text{per} A \leq \text{per} S\) for all \(S \in \Omega_ n(Z)\). Let \(C\) be the \(n\times n\) (0,1) matrix with 0's in positions \(Z\) and 1's elsewhere. \(B = (b_{ij}) \in \Omega_ n (Z)\), where \(b_{ij} = \text{per} C (i | j)/ \text{per} C\) for all \((i,j) \notin Z\), is called the barycenter of \(\Omega_ n(Z)\); if \(B\) is a minimizing matrix, \(\Omega_ n(Z)\) is said to be barycentric. It is shown that \(\Omega_ n (Z_ 2)\) and \(\Omega_ n (Z_ 3)\) are not barycentric. For a fixed \(k \geq 2\), the set \(\Omega_ n (Z_ k)\) is not barycentric, except possibly for less than \(k^ 2 + k\) values of \(n\). \(Z_ k\) is the particular \(Z = \{(1,1,), \dots,(k,k)\}\).
Mathematics Subject Classification ID: 15A15
Mathematics Subject Classification ID: 15B51
zbMATH DE Number: 614177
zbMATH Keywords: minimum permanents
zbMATH Keywords: precribed zero entries
zbMATH Keywords: doubly stochastic matrices
zbMATH Keywords: (0,1) matrix
zbMATH Keywords: barycenter
reviewed by: Richard Sinkhorn
MaRDI profile type: Publication
full work available at URL: 0024-3795(94)90111-2
OpenAlex ID: W2086556925
cites work: An interesting face of the polytope of doubly stochastic matrices
cites work: Q5672631
cites work: Proof of the van der Waerden conjecture regarding the permanent of a doubly stochastic matrix
cites work: Minimum permanents of doubly stochastic matrices with at least one zero entry†
cites work: On the permanent of doubly stochastic matrices with zero diagonal
cites work: Minimum permanents of doubly stochastic matrices with prescribed zero entries†
cites work: Theory of permanents 1982–1985
author: Henryk Minc
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description: scientific article; zbMATH DE number 614177