Injective optimal realizations of finite metric spaces

A realization of a finite metric space (X,d)(X,d) is a weighted graph (G,w)(G,w) whose vertex set contains XX such that the distances between the elements of XX in GG correspond to those given by dd. Such a realization is called optimal if it has minimal total edge weight. Optimal realizations have applications in fields such as phylogenetics, psychology, compression software and internet tomography. Given an optimal realization (G,w)(G,w) of (X,d)(X,d), there always exist certain “proper” maps from the vertex set of GG into the so-called tight span of dd. In [A. Dress, Trees, tight extensions of metric spaces, and the cohomological dimension of certain groups: a note on combinatorial properties of metric spaces, Adv. Math. 53 (1984) 321–402], Dress conjectured that any such map must be injective. Although this conjecture was recently disproven, in this paper we show that it is possible to characterize those optimal realizations (G,w)(G,w) for which certain generalizations of proper maps–that map the geometric realization of (G,w)(G,w) into the tight span instead of its vertex set–must always be injective. We also prove that these “injective” optimal realizations always exist, and show how they may be constructed from non-injective ones. Ultimately it is hoped that these results will contribute towards developing new ways to compute optimal realizations from tight spans.