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A realisation 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 betweenelements of $X$ considered as vertices in $G$ is equal to $d$. Such arealisation $(G,w)$ is called optimal if the sum of its edge weights is minimalover all such realisations. Optimal realisations always exist, although it isNP-hard to compute them in general, and they have applications in areas such asphylogenetics, electrical networks and internet tomography. In [Adv. in Math.53, 1984, 321-402] A.~Dress showed that the optimal realisations of a metric$d$ are closely related to a certain polytopal complex that can be canonicallyassociated to $d$ called its tight-span. Moreover, he conjectured that the(weighted) graph consisting of the zero- and one-dimensional faces of thetight-span of $d$ must always contain an optimal realisation as a homeomorphicsubgraph. In this paper, we prove that this conjecture does indeed hold for acertain class of metrics, namely the class of totally"=decomposable metricswhose tight-span has dimension two. As a corollary, it follows that the minimumManhattan network problem is a special case of finding optimal realisations oftwo-dimensional totally-decomposable metrics.

Optimal realisations of two-dimensional, totally-decomposable metrics