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Connected k-Center and k-Diameter Clustering

Authors Lukas Drexler, Jan Eube, Kelin Luo, Heiko Röglin, Melanie Schmidt, Julian Wargalla

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Author Details

Lukas Drexler
  • Heinrich-Heine Universität Düsseldorf, Germany
Jan Eube
  • Universität Bonn, Germany
Kelin Luo
  • Universität Bonn, Germany
Heiko Röglin
  • Universität Bonn, Germany
Melanie Schmidt
  • Heinrich-Heine Universität Düsseldorf, Germany
Julian Wargalla
  • Heinrich-Heine Universität Düsseldorf, Germany


The authors thank anonymous reviewers of a previous draft for helpful comments and pointing out relevant related work. We thank Jürgen Kusche and Christian Sohler for raising the problem and for fruitful discussion on the modeling. We also thank Xiangyu Guo for the discussion on the algorithm design and analysis.

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Lukas Drexler, Jan Eube, Kelin Luo, Heiko Röglin, Melanie Schmidt, and Julian Wargalla. Connected k-Center and k-Diameter Clustering. In 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 261, pp. 50:1-50:20, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)


Motivated by an application from geodesy, we study the connected k-center problem and the connected k-diameter problem. These problems arise from the classical k-center and k-diameter problems by adding 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. Usually in clustering problems one assumes that the clusters are pairwise disjoint. We study this case but additionally also the case that clusters are allowed to be non-disjoint. This can help to satisfy the connectivity constraints. Our main result is an O(1)-approximation algorithm for the disjoint connected k-center and k-diameter problem for Euclidean spaces of low dimension (constant d) and for metrics with constant doubling dimension. For general metrics, we get an O(log²k)-approximation. Our algorithms work by computing a non-disjoint connected clustering first and transforming it into a disjoint connected clustering. We complement these upper bounds by several upper and lower bounds for variations and special cases of the model.

Subject Classification

ACM Subject Classification
  • Theory of computation → Facility location and clustering
  • Approximation algorithms
  • Clustering
  • Connectivity constraints


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