A $(1 + {\varepsilon})$-Embedding of Low Highway Dimension Graphs into Bounded Treewidth Graphs

TL;DR

This paper presents a probabilistic embedding of low highway dimension graphs into bounded treewidth graphs with minimal distortion, enabling efficient approximation algorithms for transportation network problems.

Contribution

It introduces a novel $(1 + \\varepsilon)$-embedding technique for low highway dimension graphs into bounded treewidth graphs, extending Talwar's embedding methods.

Findings

Embedding distorts shortest paths by at most 1 + ε in expectation

Enables quasi-polynomial time approximation schemes for TSP, Steiner Tree, Facility Location

Extends geometric embedding techniques beyond low doubling metrics

Abstract

Graphs with bounded highway dimension were introduced by Abraham et al. [SODA 2010] as a model of transportation networks. We show that any such graph can be embedded into a distribution over bounded treewidth graphs with arbitrarily small distortion. More concretely, given a weighted graph G = (V, E) of constant highway dimension, we show how to randomly compute a weighted graph H = (V, E') that distorts shortest path distances of G by at most a 1 + ${\varepsilon}$ factor in expectation, and whose treewidth is polylogarithmic in the aspect ratio of G. Our probabilistic embedding implies quasi-polynomial time approximation schemes for a number of optimization problems that naturally arise in transportation networks, including Travelling Salesman, Steiner Tree, and Facility Location. To construct our embedding for low highway dimension graphs we extend Talwar's [STOC 2004] embedding of low doubling dimension metrics into bounded treewidth graphs, which generalizes known results for Euclidean metrics. We add several non-trivial ingredients to Talwar's techniques, and in particular thoroughly analyse the structure of low highway dimension graphs. Thus we demonstrate that the geometric toolkit used for Euclidean metrics extends beyond the class of low doubling metrics.