2026-09-08T01:21:15Z https://drops.dagstuhl.de/oai

oai:drops-oai.dagstuhl.de:26461 2026-09-05T19:44:18Z ddc:004 open_access

On Tight FPT Time Approximation Algorithms for k-Clustering Problems Dai, Han Li, Shi Peng, Sijin Approximation algorithms Monotone symmetric norms Clustering Fixed parameter tractability Following recent advances in combining approximation algorithms with fixed-parameter tractability (FPT), we study FPT-time approximation algorithms for minimum-norm k-clustering problems, parameterized by the number k of open facilities. For the capacitated setting, we give a tight (3+ε)-approximation for the general-norm capacitated k-clustering problem in FPT-time parameterized by k and ε. Prior to our work, such a result was only known for the capacitated k-median problem [Cohen-Addad and Li, 2019]. As a special case, our result yields an FPT-time 3-approximation for capacitated k-center. The problem has not been studied in the FPT-time setting, with the previous best known polynomial-time approximation ratio being 9 [An et al., 2015]. In the uncapacitated setting, we consider the top-cn norm k-clustering problem, where the goal of the problem is to minimize the top-cn norm of the connection distance vector. Our main result is a tight (1 + 2/(ec) + ε)-approximation algorithm for the problem with c ∈ (1/e, 1]. (For the case c ≤ 1/e, there is a simple tight (3+ε)-approximation.) Our framework can be easily extended to give a tight (3, 1 + 2/e + ε)-bi-criteria approximation for the (k-center, k-median) problem in FPT time, improving the previous best polynomial-time (4, 8) guarantee [Soroush Alamdari and David B. Shmoys, 2017]. All results are based on a unified framework: computing a (1+ε)-approximate solution using O((k log n)/ε) facilities S via LP rounding, sampling a few client representatives R based on the solution S, guessing a few pivots from S ∪ R and some radius information on the pivots, and solving the problem using the guesses. We believe this framework can lead to further results on k-clustering problems. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Han Dai and Shi Li and Sijin Peng 2026 Is Part Of LIPIcs, Volume 374, 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.ICALP.2026.72 urn:nbn:de:0030-drops-264613 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2026.72 eng https://creativecommons.org/licenses/by/4.0/legalcode