On a class of metrics related to graph layout problems

TL;DR

This paper studies metrics from embedding points on a line with minimum distances, revealing structural properties, non-closure of their convex hull, and facet-defining inequalities related to graph layout problems.

Contribution

It provides new insights into the structure of these metrics, including convex hull properties, facet characterization, and unbounded edges, linking them to graph layout problems.

Findings

Convex hull of these metrics is generally not closed.

Certain linear inequalities define facets of the convex hull closure.

Characterization of unbounded edges of the convex hull and its closure.

Abstract

We examine the metrics that arise when a finite set of points is embedded in the real line, in such a way that the distance between each pair of points is at least 1. These metrics are closely related to some other known metrics in the literature, and also to a class of combinatorial optimization problems known as graph layout problems. We prove several results about the structure of these metrics. In particular, it is shown that their convex hull is not closed in general. We then show that certain linear inequalities define facets of the closure of the convex hull. Finally, we characterise the unbounded edges of the convex hull and of its closure.