@inproceedings{a37ca5587c7342179a6976c76361fa2d,

title = "Hybrid k-Clustering: Blending k-Median and k-Center",

abstract = "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 Rd, 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) · nO(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.",

keywords = "Euclidean space, clustering, fpt approximation, k-center, k-median",

author = "Fomin, {Fedor V.} and Golovach, {Petr A.} and Tanmay Inamdar and Saket Saurabh and Meirav Zehavi",

note = "Publisher Copyright: {\textcopyright} Fedor V. Fomin, Petr A. Golovach, Tanmay Inamdar, Saket Saurabh, and Meirav Zehavi.; 27th International Conference on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2024 and the 28th International Conference on Randomization and Computation, RANDOM 2024 ; Conference date: 28-08-2024 Through 30-08-2024",

year = "2024",

month = sep,

day = "1",

doi = "10.4230/LIPIcs.APPROX/RANDOM.2024.4",

language = "English",

series = "Leibniz International Proceedings in Informatics, LIPIcs",

publisher = "Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing",

editor = "Amit Kumar and Noga Ron-Zewi",

booktitle = "Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, APPROX/RANDOM 2024",

address = "Germany",

}