Approximation Algorithms for Clustering to Minimize the Sum of Diameters

We consider the problem of partitioning the n nodes of a complete edge weighted graph into k clusters so as to minimize the sum of the diameters of the clusters. Since the problem is NP-complete, our focus is on the development of good approximation algorithms. When edge weights satisfy the triangle inequality, we present the first approximation algorithm for the problem. The approximation algorithm yields a solution which has no more than O(k) clusters such that the sum of cluster diameters is within a factor O(ln (n/k)) of the optimal value using exactly k clusters. Our approach also permits a tradeoff among the constant terms hidden by the two big-O terms and the running time. For any fixed k, we present an approximation algorithm that produces k clusters whose total diameter is at most twice the optimal value. When the distances are not required to satisfy the triangle inequality, we show that, unless P = NP, for any ρ ≥ 1, there is no polynomial time approximation algorithm that can provide a performance guarantee of ρ even when the number of clusters is fixed at 3. We also present some results for the problem of minimizing the sum of cluster radii.