Connected k-Center and k-Diameter Clustering

Lukas Drexler; Jan Eube; Kelin Luo; Dorian Reineccius; Heiko Röglin; Melanie Schmidt; Julian Wargalla

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arXiv · 2211.02176

Connected k-Center and k-Diameter Clustering

Lukas Drexler·Jan Eube·Kelin Luo·Dorian Reineccius·Heiko Röglin·Melanie Schmidt·Julian Wargalla

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Abstract

Motivated by an application from geodesy, we introduce a novel clustering problem which is a $k$-center (or k-diameter) problem with a side constraint. For the side constraint, we are given an undirected connectivity graph $G$ on the input points, and a clustering is now only feasible if every cluster induces a connected subgraph in $G$. We call the resulting problems the connected $k$-center problem and the connected $k$-diameter problem. We prove several results on the complexity and approximability of these problems. Our main result is an $O(\log^2{k})$-approximation algorithm for the connected $k$-center and the connected $k$-diameter problem. For Euclidean metrics and metrics with constant doubling dimension, the approximation factor of this algorithm improves to $O(1)$. We also consider the special cases that the connectivity graph is a line or a tree. For the line we give optimal polynomial-time algorithms and for the case that the connectivity graph is a tree, we either give an optimal polynomial-time algorithm or a $2$-approximation algorithm for all variants of our model. We complement our upper bounds by several lower bounds.

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Lukas Drexler, Jan Eube, Kelin Luo, Dorian Reineccius, Heiko Röglin, Melanie Schmidt, Julian Wargalla. 2022-11-03. Connected k-Center and k-Diameter Clustering. https://arxiv.org/abs/2211.02176

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