2026-07-20T07:30:28Z https://drops.dagstuhl.de/oai

oai:drops-oai.dagstuhl.de:17862 2024-03-06T11:00:33Z ddc:004 open_access

FPT Constant-Approximations for Capacitated Clustering to Minimize the Sum of Cluster Radii Bandyapadhyay, Sayan Lochet, William Saurabh, Saket Clustering FPT-approximation Clustering with capacity constraints is a fundamental problem that attracted significant attention throughout the years. In this paper, we give the first FPT constant-factor approximation algorithm for the problem of clustering points in a general metric into k clusters to minimize the sum of cluster radii, subject to non-uniform hard capacity constraints (Capacitated Sum of Radii ). In particular, we give a (15+ε)-approximation algorithm that runs in 2^𝒪(k²log k) ⋅ n³ time. When capacities are uniform, we obtain the following improved approximation bounds. - A (4 + ε)-approximation with running time 2^𝒪(klog(k/ε)) n³, which significantly improves over the FPT 28-approximation of Inamdar and Varadarajan [ESA 2020]. - A (2 + ε)-approximation with running time 2^𝒪(k/ε² ⋅log(k/ε)) dn³ and a (1+ε)-approxim- ation with running time 2^𝒪(kdlog ((k/ε))) n³ in the Euclidean space. Here d is the dimension. - A (1 + ε)-approximation in the Euclidean space with running time 2^𝒪(k/ε² ⋅log(k/ε)) dn³ if we are allowed to violate the capacities by (1 + ε)-factor. We complement this result by showing that there is no (1 + ε)-approximation algorithm running in time f(k)⋅ n^𝒪(1), if any capacity violation is not allowed. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Sayan Bandyapadhyay and William Lochet and Saket Saurabh 2023 Is Part Of LIPIcs, Volume 258, 39th International Symposium on Computational Geometry (SoCG 2023) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.SoCG.2023.12 urn:nbn:de:0030-drops-178628 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2023.12 eng https://creativecommons.org/licenses/by/4.0/legalcode