Abstract
The NonUniform kcenter (NUkC) problem has recently been formulated by Chakrabarty, Goyal and Krishnaswamy [ICALP, 2016] as a generalization of the classical kcenter clustering problem. In NUkC, given a set of n points P in a metric space and nonnegative numbers r₁, r₂, … , r_k, the goal is to find the minimum dilation α and to choose k balls centered at the points of P with radius α⋅ r_i for 1 ≤ i ≤ k, such that all points of P are contained in the union of the chosen balls. They showed that the problem is NPhard to approximate within any factor even in tree metrics. On the other hand, they designed a "bicriteria" constant approximation algorithm that uses a constant times k balls. Surprisingly, no true approximation is known even in the special case when the r_i’s belong to a fixed set of size 3. In this paper, we study the NUkC problem under perturbation resilience, which was introduced by Bilu and Linial [Combinatorics, Probability and Computing, 2012]. We show that the problem under 2perturbation resilience is polynomial time solvable when the r_i’s belong to a constant sized set. However, we show that perturbation resilience does not help in the general case. In particular, our findings imply that even with perturbation resilience one cannot hope to find any "good" approximation for the problem.
BibTeX  Entry
@InProceedings{bandyapadhyay:LIPIcs:2020:12634,
author = {Sayan Bandyapadhyay},
title = {{On Perturbation Resilience of NonUniform kCenter}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020)},
pages = {31:131:22},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959771641},
ISSN = {18688969},
year = {2020},
volume = {176},
editor = {Jaros{\l}aw Byrka and Raghu Meka},
publisher = {Schloss DagstuhlLeibnizZentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2020/12634},
URN = {urn:nbn:de:0030drops126347},
doi = {10.4230/LIPIcs.APPROX/RANDOM.2020.31},
annote = {Keywords: NonUniform kcenter, stability, clustering, perturbation resilience}
}
Keywords: 

NonUniform kcenter, stability, clustering, perturbation resilience 
Collection: 

Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020) 
Issue Date: 

2020 
Date of publication: 

11.08.2020 