Optimal realizations of two-dimensional, totally-decomposable metrics

Herrmann, SvenUniv E Anglia, Sch Comp Sci, Norwich NR4 7TJ, Norfolk, England. (author)

A realization of a metric d on a finite set X is a weighted graph (G, w) whose vertex set contains X such that the shortest-path distance between elements of X considered as vertices in G is equal to d. Such a realization (G, w) 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 dare closely related to a certain polytopal complex that can be canonically associated to d called its tight-span. Moreover, he conjectured that the (weighted) graph consisting of the zero- and one-dimensional faces of the tight-span of d 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. (C) 2015 Elsevier B.V. All rights reserved.

Koolen, Jack H.Univ Sci & Technol China, Sch Math Sci, Wen Tsun Wu Key Lab, CAS, Hefei, Peoples R China. (author)
Lesser, AliceUppsala universitet,Matematiska institutionen(Swepub:uu)alles844 (author)
Moulton, VincentUniv E Anglia, Sch Comp Sci, Norwich NR4 7TJ, Norfolk, England. (author)
Wu, TaoyangUniv E Anglia, Sch Comp Sci, Norwich NR4 7TJ, Norfolk, England.;Natl Univ Singapore, Dept Math, Singapore 119076, Singapore. (author)
Univ E Anglia, Sch Comp Sci, Norwich NR4 7TJ, Norfolk, England.Univ Sci & Technol China, Sch Math Sci, Wen Tsun Wu Key Lab, CAS, Hefei, Peoples R China. (creator_code:org_t)

By the author/editor: Herrmann, Sven; Koolen, Jack H.; Lesser, Alice; Moulton, Vincent; Wu, Taoyang

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NATURAL SCIENCES: NATURAL SCIENCES; and Mathematics; and Discrete Mathema ...

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