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

We study the LowerBoundedCenter (LBC) problem, which is a clustering problem that can be viewed as a variant of the k-Center problem. In the LBC problem, we are given a set of points P in a metric space and a lower bound λ, and the goal is to select a set C ⊆ 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 λ 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 IR, where d is a constant. We also prove that this problem cannot be approximated within a factor of 1.8− unless P = NP even if the input points are points in the Euclidean plane IR.