# Hierarchy of Transportation Network Parameters and Hardness Results

## TL;DR

This paper explores the relationships between transportation network graph parameters, establishes their computational hardness, and demonstrates the complexity of related optimization problems on graphs with bounded parameters.

## Contribution

It analyzes the relationships between highway and skeleton dimensions, proves NP-hardness of computing and approximating problems on such graphs, and clarifies their theoretical properties.

## Key findings

- Skeleton dimension is incomparable to several other parameters.
- Computing highway dimension is NP-hard.
- Approximating k-Center within factor less than 2 is NP-hard on certain graphs.

## Abstract

The graph parameters highway dimension and skeleton dimension were introduced to capture the properties of transportation networks. As many important optimization problems like Travelling Salesperson, Steiner Tree or $k$-Center arise in such networks, it is worthwhile to study them on graphs of bounded highway or skeleton dimension. We investigate the relationships between mentioned parameters and how they are related to other important graph parameters that have been applied successfully to various optimization problems. We show that the skeleton dimension is incomparable to any of the parameters distance to linear forest, bandwidth, treewidth and highway dimension and hence, it is worthwhile to study mentioned problems also on graphs of bounded skeleton dimension. Moreover, we prove that the skeleton dimension is upper bounded by the max leaf number and that for any graph on at least three vertices there are edge weights such that both parameters are equal. Then we show that computing the highway dimension according to most recent definition is NP-hard, which answers an open question stated by Feldmann et al. Finally we prove that on graphs $G=(V,E)$ of skeleton dimension $\mathcal{O}(\log^2 \vert V \vert)$ it is NP-hard to approximate the $k$-Center problem within a factor less than $2$.