Searching for Realizations of Finite Metric Spaces in Tight Spans

TL;DR

This paper introduces a new heuristic algorithm for finding minimal edge-weighted graph representations of finite metrics by leveraging the tight span structure, offering an alternative to existing methods and demonstrating promising computational results.

Contribution

The paper presents a novel heuristic that exploits the tight span of a metric to find graph realizations, improving upon previous approaches and applicable to specific metric types.

Findings

Effective for $l_1$-distances and two-decomposable metrics

Comparable or improved performance over existing heuristics

Provides computational evidence of efficiency and accuracy

Abstract

An important problem that commonly arises in areas such as internet traffic-flow analysis, phylogenetics and electrical circuit design, is to find a representation of any given metric $D$ on a finite set by an edge-weighted graph, such that the total edge length of the graph is minimum over all such graphs. Such a graph is called an optimal realization and finding such realizations is known to be NP-hard. Recently Varone presented a heuristic greedy algorithm for computing optimal realizations. Here we present an alternative heuristic that exploits the relationship between realizations of the metric $D$ and its so-called tight span $T_D$. The tight span $T_D$ is a canonical polytopal complex that can be associated to $D$, and our approach explores parts of $T_D$ for realizations in a way that is similar to the classical simplex algorithm. We also provide computational results illustrating the performance of our approach for different types of metrics, including $l_1$-distances and two-decomposable metrics for which it is provably possible to find optimal realizations in their tight spans.