Minimum sum of diameters clustering

The problem of determining a partition of a given set ofN entities intoM clusters such that the sum of the diameters of these clusters is minimum has been studied by Brucker (1978). He proved that it is NP-complete forM3 and mentioned that its complexity was unknown forM=2. We provide anO(N ³ logN) algorithm for this latter case. Moreover, we show that determining a partition into two clusters which minimizes any given function of the diameters can be done inO(N ⁵) time.Acknowledgments: This research was supported by the Air Force Office of Scientific Research Grant AFOSR 0271 to Rutgers University. We are grateful to Yves Crama for several insightful remarks and to an anonymous referee for detailed comments.