[Paper Review] On tight spans and tropical polytopes for directed distances

Hiroshi Hirai, Shungo Koichi|arXiv (Cornell University)|Apr 3, 2010

Polynomial and algebraic computation14 references3 citations

TL;DR

This paper develops a directed analogue of the tight span and tropical polytope for directed metrics, establishing universal embedding properties for tight and cyclically tight extensions. It proves a directed tree metric theorem: a directed metric is a directed tree metric if and only if the tropical rank of its negated distance matrix is at most two.

ABSTRACT

An extension $(V,d)$ of a metric space $(S,μ)$ is a metric space with $S \subseteq V$ and $d|_S = μ$, and is said to be tight if there is no other extension $(V,d')$ of $(S,μ)$ with $d' \leq d$. Isbell and Dress independently found that every tight extension is isometrically embedded 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 such a universal embedding property for tight extensions. Also we newly introduce another natural class of extensions, called cyclically tight extensions, and show that (a fiber of) the tropical polytope, introduced by Develin and Sturmfels, has a universal embedding property for cyclically tight extensions. As an application, we prove the following directed version of tree metric theorem: 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 in multicommodity flows in directed networks.

Motivation & Objective

• To extend the theory of tight spans and tropical polytopes to directed metrics, which are not necessarily symmetric.
• To define and characterize tight and cyclically tight extensions in the context of directed distances.
• To establish universal embedding properties for these extensions using directed analogues of the tight span and tropical polytope.
• To apply the framework to multicommodity flow problems in directed networks, linking geometry to optimization.
• To provide a directed version of the classical tree metric theorem using tropical rank.

Proposed method

• Define the directed tight span $T_{ ho}$ as the set of minimal points in the unbounded polyhedron $P_{ ho} = \{p \in \mathbb{R}^S \mid p(s) + p(t) \geq \rho(s,t)\}$.
• Introduce the tropical polytope $\bar{Q}_{\rho}$ as the projection of the set of minimal points in another polyhedron $Q_{\rho}$, and define a balanced section $R \subseteq Q_{\rho}$ that projects bijectively to $\bar{Q}_{\rho}$.
• Equip $T_{\rho}$, $Q_{\rho}$, and any balanced section $R$ with a directed $l_\infty$-metric $D_\infty$.
• Prove that every tight extension of a directed metric $\rho$ is isometrically embedded into $(T_{\rho}, D_\infty)$, and every cyclically tight extension is isometrically embedded into a balanced section of $Q_{\rho}$.
• Use the dimension criteria of $T_{\rho}$ and $\bar{Q}_{\rho}$ to characterize directed tree metrics.
• Apply the framework to multicommodity flow problems by showing that the dual linear program reduces to facility location problems on $T_{\rho}$ and $\bar{Q}_{\rho}$.

Experimental results

Research questions

• RQ1 Does a universal tight extension exist for directed metrics, analogous to the tight span in the symmetric case?
• RQ2 Can a directed analogue of the tropical polytope be constructed that universally embeds cyclically tight extensions?
• RQ3 What combinatorial characterization exists for directed tree metrics in terms of tropical geometry?
• RQ4 How can the geometry of directed tight spans and tropical polytopes be applied to multicommodity flow problems in directed networks?
• RQ5 What is the relationship between the tropical rank of $-\mu$ and the structure of directed tree metrics?

Key findings

• The directed tight span $T_{\rho}$ is the universal tight extension for a directed metric $\rho$, meaning every tight extension embeds isometrically into $T_{\rho}$ with the $D_\infty$ metric.
• The tropical polytope $\bar{Q}_{\rho}$, via a balanced section, provides a universal embedding for cyclically tight extensions of $\rho$.
• A directed metric $\rho$ is a directed tree metric if and only if the tropical rank of $-\rho$ is at most two, establishing a directed version of the tree metric theorem.
• The dual of the multicommodity flow problem in a general network reduces to a facility location problem on the directed tight span $T_{\rho}$, with the objective function expressed as a sum of $D_\infty$-distances over edges.
• For Eulerian networks, the dual problem reduces to a facility location problem on the tropical polytope $\bar{Q}_{\rho}$, with the objective function minimized over embeddings into any balanced section of $Q_{\rho}$.
• The framework provides a geometric interpretation of multicommodity flow duality, linking optimization to metric geometry in directed spaces.