W[1]-Hardness of the k-Center Problem Parameterized by the Skeleton Dimension

TL;DR

This paper proves that the k-Center problem remains computationally hard (W[1]-hard) even on graphs with low skeleton dimension, and establishes strong complexity lower bounds under the Exponential Time Hypothesis.

Contribution

It extends previous hardness results for k-Center to include skeleton dimension as a parameter, showing persistent complexity even in restricted graph classes.

Findings

k-Center is W[1]-hard on graphs with low skeleton dimension

No efficient exact algorithm exists under ETH with certain runtime bounds

Hardness persists even when parameterized by multiple graph parameters

Abstract

In the $k$-Center problem, we are given a graph $G=(V,E)$ with positive edge weights and an integer $k$ and the goal is to select $k$ center vertices $C \subseteq V$ such that the maximum distance from any vertex to the closest center vertex is minimized. On general graphs, the problem is NP-hard and cannot be approximated within a factor less than $2$. Typical applications of the $k$-Center problem can be found in logistics or urban planning and hence, it is natural to study the problem on transportation networks. Such networks are often characterized as graphs that are (almost) planar or have low doubling dimension, highway dimension or skeleton dimension. It was shown by Feldmann and Marx that $k$-Center is W[1]-hard on planar graphs of constant doubling dimension when parameterized by the number of centers $k$, the highway dimension $hd$ and the pathwidth $pw$. We extend their result and show that even if we additionally parameterize by the skeleton dimension $\kappa$, the $k$-Center problem remains W[1]-hard. Moreover, we prove that under the Exponential Time Hypothesis there is no exact algorithm for $k$-Center that has runtime $f(k,hd,pw,\kappa) \cdot \vert V \vert^{o(pw + \kappa + \sqrt{k+hd})}$ for any computable function $f$.