We propose a novel clustering model encompassing two well-known clustering models: k-center clustering and k-median clustering. In the Hybrid k-Clustering problem, given a set P of points in ℝ^d, an integer k, and a non-negative real r, our objective is to position k closed balls of radius r to minimize the sum of distances from points not covered by the balls to their closest balls. Equivalently, we seek an optimal L₁-fitting of a union of k balls of radius r to a set of points in the Euclidean space. When r = 0, this corresponds to k-median; when the minimum sum is zero, indicating complete coverage of all points, it is k-center. Our primary result is a bicriteria approximation algorithm that, for a given ε > 0, produces a hybrid k-clustering with balls of radius (1+ε)r. This algorithm achieves a cost at most 1+ε of the optimum, and it operates in time 2^{(kd/ε)^𝒪(1)} ⋅ n^𝒪(1). Notably, considering the established lower bounds on k-center and k-median, our bicriteria approximation stands as the best possible result for Hybrid k-Clustering.
@InProceedings{fomin_et_al:LIPIcs.APPROX/RANDOM.2024.4, author = {Fomin, Fedor V. and Golovach, Petr A. and Inamdar, Tanmay and Saurabh, Saket and Zehavi, Meirav}, title = {{Hybrid k-Clustering: Blending k-Median and k-Center}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024)}, pages = {4:1--4:19}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-348-5}, ISSN = {1868-8969}, year = {2024}, volume = {317}, editor = {Kumar, Amit and Ron-Zewi, Noga}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2024.4}, URN = {urn:nbn:de:0030-drops-209975}, doi = {10.4230/LIPIcs.APPROX/RANDOM.2024.4}, annote = {Keywords: clustering, k-center, k-median, Euclidean space, fpt approximation} }
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