Abstract
Classical clustering problems search for a partition of objects into a fixed number of clusters. In many scenarios, however, the number of clusters is not known or necessarily fixed. Further, clusters are sometimes only considered to be of significance if they have a certain size. We discuss clustering into sets of minimum cardinality k without a fixed number of sets and present a general model for these types of problems. This general framework allows the comparison of different measures to assess the quality of a clustering. We specifically consider nine quality-measures and classify the complexity of the resulting problems with respect to k. Further, we derive some polynomial-time solvable cases for \(k=2\) with connections to matching-type problems which, among other graph problems, then are used to compute approximations for larger values of k.
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Notes
This covering problem is sometimes also called Unweighted Simplex Matching and is equivalent to \(\{K_2,K_3\}\)-packing, an old, well studied generalisation of the classical matching problem [7].
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Acknowledgements
Katrin Casel and Henning Fernau were supported by the German Science Foundation Deutsche Forschungsgemeinschaft (FE 560/6-1). Faisal Abu-Khzam and Cristina Bazgan were partially supported by the bilateral research cooperation CEDRE between France and Lebanon (Grant Number 30885TM). We are grateful for the helpful comments of the anonymous reviewers.
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Abu-Khzam, F.N., Bazgan, C., Casel, K. et al. Clustering with Lower-Bounded Sizes. Algorithmica 80, 2517–2550 (2018). https://doi.org/10.1007/s00453-017-0374-5
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DOI: https://doi.org/10.1007/s00453-017-0374-5