Highway Dimension: a Metric View

TL;DR

This paper generalizes the concept of highway dimension to include more metric spaces like grids and Euclidean planes, and develops algorithms and tools for these spaces, improving approximation schemes for problems like TSP.

Contribution

It introduces a relaxed highway dimension definition that encompasses more metric spaces and provides a PTAS for TSP, along with foundational metric tools for such spaces.

Findings

Generalized highway dimension includes grid and Euclidean spaces.

Developed a PTAS for TSP under the new highway dimension definition.

Constructed metric tools like padded decompositions and sparse covers.

Abstract

Realistic metric spaces (such as road/transportation networks) tend to be much more algorithmically tractable than general metrics. In an attempt to formalize this intuition, Abraham et~al.\ (SODA 2010, JACM 2016) introduced the notion of highway dimension. A weighted graph $G$ has highway dimension $h$ if for every ball $B$ of radius $\approx4r$, there is a hitting set of size $h$ hitting all the shortest paths of length $>r$ in $B$. Unfortunately, this definition fails to incorporate some very natural metric spaces such as the grid graph, and the Euclidean plane. We relax the definition of highway dimension by demanding to hit only approximate shortest paths. In addition to generalizing the original definition, this new definition also incorporates all doubling spaces (in particular the grid graph and the Euclidean plane). We then construct a PTAS for TSP under this new definition (improving a QPTAS w.r.t.\ the original more restrictive definition of Feldmann et~al.\ (SICOMP 2018)). Finally, we develop a basic metric toolkit for spaces with small highway dimension by constructing padded decompositions, sparse covers/partitions, and tree covers. An abundance of applications follow.