Maximizing the ratio of cluster split to cluster diameter without and with cardinality constraints.

k -Clustering partitions a set of n points in a metric space into at most k clusters in a way that makes the resulted clusters satisfy both the homogeneity and the separation criteria. The diameter of a cluster is the maximum distance between pairs of points in the cluster, and the split of a cluster is the minimum distance between points of the cluster and ones outside the cluster. In this paper, we propose a new criterion for measuring the goodness of clusters: The ratio of the minimum cluster split to the maximum cluster diameter (abbr. RSD). We study the following two optimization problems: Maximizing RSD without and with cardinality constraints, i.e., each cluster should consist of at least N points. For both problems with k ≥ 3 , it is NP-hard to achieve a factor of (1 2 + ϵ) approximation for any ϵ > 0. For k = 2 , we solve the two problems optimally; For k ≥ 3 , we present a 1 2 -approximation and a 1 4 -approximation algorithm for the two problems, respectively. [ABSTRACT FROM AUTHOR]