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Connected k-Median with Disjoint and Non-Disjoint Clusters

Authors Jan Eube , Kelin Luo , Dorian Reineccius , Heiko Röglin , Melanie Schmidt

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

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Abstract

The connected k-median problem is a constrained clustering problem that combines distance-based k-clustering with connectivity information. The problem allows to input a metric space and an unweighted undirected connectivity graph that is completely unrelated to the metric space. The goal is to compute k centers and corresponding clusters such that each cluster forms a connected subgraph of G, and such that the k-median cost is minimized.
The problem has applications in very different fields like geodesy (particularly districting), social network analysis (especially community detection), or bioinformatics. We study a version with overlapping clusters where points can be part of multiple clusters which is natural for the use case of community detection. This problem variant is Ω(log n)-hard to approximate, and our main result is an 𝒪(k² log n)-approximation algorithm for the problem. We complement it with an Ω(n^{1-ε})-hardness result for the case of disjoint clusters without overlap with general connectivity graphs, as well as an exact algorithm in this setting if the connectivity graph is a tree.

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Jan Eube, Kelin Luo, Dorian Reineccius, Heiko Röglin, and Melanie Schmidt. Connected k-Median with Disjoint and Non-Disjoint Clusters. In 33rd Annual European Symposium on Algorithms (ESA 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 351, pp. 63:1-63:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.ESA.2025.63

Author Details

Jan Eube
  • University of Bonn, Germany
Kelin Luo
  • University at Buffalo, NY, USA
Dorian Reineccius
  • Deutsches Zentrum für Luft- und Raumfahrt, Bonn, Germany
Heiko Röglin
  • University of Bonn, Germany
Melanie Schmidt
  • Heinrich-Heine Universität Düsseldorf, Germany

Funding

This work has been funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) – 390685813; 459420781 and by the Lamarr Institute for Machine Learning and Artificial Intelligence lamarr-institute.org.

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