Geometric clustering

Abstract

We study the parameterized complexity of the k -center problem on a given n -point set P in ℝ d , with the dimension d as the parameter. We show that the rectilinear 3-center problem is fixed-parameter tractable, by giving an algorithm that runs in O ( n log n ) time for any fixed dimension d . On the other hand, we show that this is unlikely to be the case with both the Euclidean and rectilinear k -center problems for any k ≥ 2 and k ≥ 4 respectively. In particular, we prove that deciding whether P can be covered by the union of 2 balls of given radius or by the union of 4 cubes of given side length is W[1]-hard with respect to d , and thus not fixed-parameter tractable unless FPT=W[1]. For the Euclidean case, we also show that even an n o ( d ) -time algorithm does not exist, unless there is a 2 o ( n ) -time algorithm for n -variable 3SAT, that is, the Exponential Time Hypothesis fails.