Polynomial time approximation schemes for clustering in low highway dimension graphs

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.

Generalized $$k$$-Center: Distinguishing Doubling and Highway Dimension

“…For the k-Median problem parameterized by the doubling dimension, Cohen-Addad et al [13] show an efficient parameterized approximation scheme (EPAS), which is a parameterized algorithm that for some parameter q and any ε > 0 outputs a solution of cost at most (1 + ε)OPT and runs in time O * (f (q, ε)) where f is a computable function. In graphs of constant highway dimension, Feldmann and Saulpic [24] follow up with a polynomial time approximation scheme (PTAS) for k-Median. If we allow k as a parameter as well, then Feldmann and Marx [23] show an EPAS for k-Center in low doubling dimension graphs, while Becker et al [7] show an EPAS for k-Center in low highway dimension graphs.…”