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., 2018) [8] or run in FPT time (parameterized by the number of clusters k, the highway dimension, and the approximation factor) (Becker et al., 2018; Braverman et al., 2021) [9,10]. 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.
Highlights
Clustering is a standard optimization task that seeks a “good” partition of a metric space, such that two points that are “close” should be in the same part
The metric space can be given by a transportation network, which can be modeled by graphs with low highway dimension
In this work we focus on transportation networks, for which it can be argued that metric spaces with bounded doubling dimension are not a suitable model: for instance, hub-andspoke networks seen in air traffic networks do not have low doubling dimension
Summary
Clustering is a standard optimization task that seeks a “good” partition of a metric space, such that two points that are “close” should be in the same part. The highway dimension of a graph G is the smallest integer h such that, for some universal constant c > 4, for every r ∈ R+, and every ball βv(cr) of radius cr, there are at most h vertices in βv(cr) hitting all shortest paths of length more than r that lie in βv(cr) For this class of graphs, the only known approximation algorithms for clustering that compute (1 + ε)-approximations for any ε > 0 either run in quasi-polynomial time, i.e., QPTASs [14], or with runtime f (h, k, ε) · n for some exponential function f , i.e., parameterized approximation schemes [6, 8]. An open problem is to identify polynomial-time approximation schemes (PTASs) for clustering in graphs of constant highway dimension
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