We give a constant factor polynomial time pseudo-approximation algorithm for min-sum clustering with or without outliers. The algorithm is allowed to exclude an arbitrarily small constant fraction of the points. For instance, we show how to compute a solution that clusters 98% of the input data points and pays no more than a constant factor times the optimal solution that clusters 99% of the input data points. More generally, we give the following bicriteria approximation: For any ε > 0, for any instance with n input points and for any positive integer n' ≤ n, we compute in polynomial time a clustering of at least (1-ε) n' points of cost at most a constant factor greater than the optimal cost of clustering n' points. The approximation guarantee grows with 1/(ε). Our results apply to instances of points in real space endowed with squared Euclidean distance, as well as to points in a metric space, where the number of clusters, and also the dimension if relevant, is arbitrary (part of the input, not an absolute constant).
@InProceedings{banerjee_et_al:LIPIcs.APPROX/RANDOM.2021.16, author = {Banerjee, Sandip and Ostrovsky, Rafail and Rabani, Yuval}, title = {{Min-Sum Clustering (With Outliers)}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021)}, pages = {16:1--16:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-207-5}, ISSN = {1868-8969}, year = {2021}, volume = {207}, editor = {Wootters, Mary and Sanit\`{a}, Laura}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2021.16}, URN = {urn:nbn:de:0030-drops-147093}, doi = {10.4230/LIPIcs.APPROX/RANDOM.2021.16}, annote = {Keywords: Clustering, approximation algorithms, primal-dual} }
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