# Travelling on Graphs with Small Highway Dimension

## TL;DR

This paper investigates the complexity and approximation algorithms for TSP and Steiner Tree problems on graphs with low highway dimension, providing an FPTAS for highway dimension 1 and establishing NP-hardness results.

## Contribution

It introduces an FPTAS for TSP on graphs with highway dimension 1 and proves NP-hardness for TSP on graphs with highway dimension 6, advancing understanding of these problems in transportation network models.

## Key findings

- FPTAS developed for TSP when highway dimension is 1
- NP-hardness of Steiner Tree in low highway dimension graphs
- NP-hardness of TSP for highway dimension 6

## Abstract

We study the Travelling Salesperson (TSP) and the Steiner Tree problem (STP) in graphs of low highway dimension. This graph parameter was introduced by Abraham et al. [SODA 2010] as a model for transportation networks, on which TSP and STP naturally occur for various applications in logistics. It was previously shown [Feldmann et al. ICALP 2015] that these problems admit a quasi-polynomial time approximation scheme (QPTAS) on graphs of constant highway dimension. We demonstrate that a significant improvement is possible in the special case when the highway dimension is 1, for which we present a fully-polynomial time approximation scheme (FPTAS). We also prove that STP is weakly NP-hard for these restricted graphs. For TSP we show NP-hardness for graphs of highway dimension 6, which answers an open problem posed in [Feldmann et al. ICALP 2015].