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Issue RAIRO Oper. Res.
Volume 41, Number 4, October-December 2007
Page(s) 361 - 366
DOI 10.1051/ro:2007028
Published online 11 October 2007

RAIRO Oper. Res. 41 (2007) 361-366
DOI: 10.1051/ro:2007028

A note on tree realizations of matrices

Alain Hertz1 and Sacha Varone2

1  Département de mathématiques et de génie industriel, École Polytechnique, Montréal, Canada; alain.hertz@gerad.ca
2  Haute école de gestion de Genève, Économie d'Entreprise, Genève, Switzerland; sacha.varone@hesge.ch

(Received March 07, 2005. Accepted January 19, 2007. Published online 11 October 2007.)

Abstract
It is well known that each tree metric M has a unique realization as a tree, and that this realization minimizes the total length of the edges among all other realizations of M. We extend this result to the class of symmetric matrices M with zero diagonal, positive entries, and such that $ m_{ij}+m_{kl} \leq max \{ m_{ik}+m_{jl},m_{il}+m_{jk}\}$ for all distinct i,j,k,l.


Mathematics Subject Classification. 05C50, 05B20, 68R10, 68U99

Key words: Matrices, tree metrics, 4-point condition


© EDP Sciences, ROADEF, SMAI 2007


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