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arXiv · 1304.7318

Fast Clustering with Lower Bounds: No Customer too Far, No Shop too Small

Alina Ene·Sariel Har-Peled·Benjamin Raichel
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Abstract

We study the \LowerBoundedCenter (\lbc) problem, which is a clustering problem that can be viewed as a variant of the \kCenter problem. In the \lbc problem, we are given a set of points P in a metric space and a lower bound \lambda, and the goal is to select a set C \subseteq P of centers and an assignment that maps each point in P to a center of C such that each center of C is assigned at least \lambda points. The price of an assignment is the maximum distance between a point and the center it is assigned to, and the goal is to find a set of centers and an assignment of minimum price. We give a constant factor approximation algorithm for the \lbc problem that runs in O(n \log n) time when the input points lie in the d-dimensional Euclidean space R^d, where d is a constant. We also prove that this problem cannot be approximated within a factor of 1.8-\epsilon unless P = \NP even if the input points are points in the Euclidean plane R^2.

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Alina Ene, Sariel Har-Peled, Benjamin Raichel. 2013-04-27. Fast Clustering with Lower Bounds: No Customer too Far, No Shop too Small. https://arxiv.org/abs/1304.7318

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