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 + epsilon) approximation for any epsilon > 0. For k = 2, we solve the two problems optimally; For k >= 3, we present a 1/2-approximation and a 1/2-approximation algorithm for the two problems, respectively. (C) 2021 Elsevier B.V. All rights reserved.