Abstract
It is well known that some local search heuristics for -clustering problems, such as -means heuristic for minimum sum-of-squares clustering occasionally stop at a solution with a smaller number of clusters than the desired number . Such solutions are called degenerate. In this paper, we reveal that the degeneracy also exists in -harmonic means (KHM) method, proposed as an alternative to -means heuristic, but which is less sensitive to the initial solution. In addition, we discover two types of degenerate solutions and provide examples for both. Based on these findings, we give a simple method to remove degeneracy during the execution of the KHM heuristic; it can be used as a part of any other heuristic for KHM clustering problem. We use KHM heuristic within a recent variant of variable neighborhood search (VNS) based heuristic. Extensive computational analysis, performed on test instances usually used in the literature, shows that significant improvements are obtained if our simple degeneracy correcting method is used within both KHM and VNS. Moreover, our VNS based heuristic suggested here may be considered as a new state-of-the-art heuristic for solving KHM clustering problem.