On Tight Spans for Directed Distances

Abstract

An extension (V, d) of a metric space (S, μ) is a metric space with ${{S \subseteq V}}$ and ${{d\mid{_S} = \mu}}$ , and is said to be tight if there is no other extension (V, d′) of (S, μ) with d′ ≤ d . Isbell and Dress independently found that every tight extension embeds isometrically into a certain metrized polyhedral complex associated with (S, μ), called the tight span. This paper develops an analogous theory for directed metrics, which are “not necessarily symmetric” distance functions satisfying the triangle inequality. We introduce a directed version of the tight span and show that it has a universal embedding property for tight extensions. Also we introduce a new natural class of extensions, called cyclically tight extensions, and we show that there also exists a certain polyhedral complex having a universal property relative to cyclically tightness. This polyhedral complex coincides with (a fiber of) the tropical polytope spanned by the column vectors of –μ, which was earlier introduced by Develin and Sturmfels. Thus this gives a tight-span interpretation to the tropical polytope generated by a nonnegative square matrix satisfying the triangle inequality. As an application, we prove the following directed version of the tree metric theorem: A directed metric μ is a directed tree metric if and only if the tropical rank of –μ is at most two. Also we describe how tight spans and tropical polytopes are applied to the study of multicommodity flows in directed networks.