Optimal realisations of two-dimensional, totally split-decomposable metrics

Herrmann, Sven, Koolen, Jack H., Lesser, Alice, Moulton, Vincent ORCID: https://orcid.org/0000-0001-9371-6435 and Wu, Taoyang ORCID: https://orcid.org/0000-0002-2663-2001 (2015) Optimal realisations of two-dimensional, totally split-decomposable metrics. Discrete Mathematics, 338 (8). 1289–1299. ISSN 0012-365X

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Abstract

A realization of a metric on a finite set is a weighted graph whose vertex set contains such that the shortest-path distance between elements of considered as vertices in is equal to . Such a realization is called optimal if the sum of its edge weights is minimal over all such realizations. Optimal realizations always exist, although it is NP-hard to compute them in general, and they have applications in areas such as phylogenetics, electrical networks and internet tomography. A. Dress (1984) showed that the optimal realizations of a metric are closely related to a certain polytopal complex that can be canonically associated to called its tight-span. Moreover, he conjectured that the (weighted) graph consisting of the zero- and one-dimensional faces of the tight-span of must always contain an optimal realization as a homeomorphic subgraph. In this paper, we prove that this conjecture does indeed hold for a certain class of metrics, namely the class of totally-decomposable metrics whose tight-span has dimension two. As a corollary, it follows that the minimum Manhattan network problem is a special case of finding optimal realizations of two-dimensional totally-decomposable metrics.

Item Type: Article
Faculty \ School: Faculty of Science > School of Computing Sciences
Depositing User: Pure Connector
Date Deposited: 22 Apr 2015 11:28
Last Modified: 21 Oct 2022 00:38
URI: https://ueaeprints.uea.ac.uk/id/eprint/53242
DOI: 10.1016/j.disc.2015.02.008

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