We present a linear time algorithm for the weighted k-center problem on trees for fixed k. This partially settles the long-standing question about the lower bound on the time complexity of the problem. The current time complexity of the best-known algorithm for the problem with k as part of the input is O(n log n) by Wang et al. [Haitao Wang and Jingru Zhang, 2018]. Whether an O(n) time algorithm exists for arbitrary k is still open.
@InProceedings{bhattacharya_et_al:LIPIcs.ISAAC.2019.27, author = {Bhattacharya, Binay and Das, Sandip and Dev, Subhadeep Ranjan}, title = {{The Weighted k-Center Problem in Trees for Fixed k}}, booktitle = {30th International Symposium on Algorithms and Computation (ISAAC 2019)}, pages = {27:1--27:11}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-130-6}, ISSN = {1868-8969}, year = {2019}, volume = {149}, editor = {Lu, Pinyan and Zhang, Guochuan}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2019.27}, URN = {urn:nbn:de:0030-drops-115238}, doi = {10.4230/LIPIcs.ISAAC.2019.27}, annote = {Keywords: facility location, prune and search, parametric search, k-center problem, conditional k-center problem, trees} }
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