Polynomial Time Approximation Schemes for Clustering in Low Highway Dimension Graphs

TL;DR

This paper proves the existence of polynomial-time approximation schemes for clustering problems like k-Median, k-Means, and Facility Location in graphs with low highway dimension, and establishes NP-hardness even in graphs of highway dimension 1.

Contribution

It introduces a PTAS for clustering in low highway dimension graphs and proves NP-hardness in the simplest case of highway dimension 1.

Findings

PTAS exists for clustering in low highway dimension graphs

NP-hardness shown for graphs with highway dimension 1

Improves understanding of computational complexity in transportation network models

Abstract

We study clustering problems such as k-Median, k-Means, and Facility Location in graphs of low highway dimension, which is a graph parameter modeling transportation networks. It was previously shown that approximation schemes for these problems exist, which either run in quasi-polynomial time (assuming constant highway dimension) [Feldmann et al. SICOMP 2018] or run in FPT time (parameterized by the number of clusters $k$, the highway dimension, and the approximation factor) [Becker et al. ESA~2018, Braverman et al. 2020]. In this paper we show that a polynomial-time approximation scheme (PTAS) exists (assuming constant highway dimension). We also show that the considered problems are NP-hard on graphs of highway dimension 1.