Optimal realizations of two-dimensional, totally-decomposable metrics

Abstract

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 between elements of X considered as vertices in G is equal to d. Such a realisation (G, w) is called optimal if the sum of its edge weights is minimal over all such realisations. Optimal realisations 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. 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 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 realisation 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 realisations of two-dimensional totally-decomposable metrics.

Searching for realizations of finite metric spaces in tight spans

“…This allows us to obtain some impression of how close the realizations computed by our heuristic are to a minimal subrealization of G D in case |X| is not too large. Moreover, in case the metric is two-decomposable, a minimal subrealization of G D is (by the aforementioned result in [20]) an optimal realization and so we can compare the realizations computed by our new heuristic with optimal ones for this special class of metrics. Based on these considerations, in Section 7 we present simulations for l 1 -distances, two-decomposable metrics and random metrics to assess the performance of our heuristic.…”