Size Constrained Distance Clustering: Separation Properties and Some Complexity Results
In this paper we study the complexity of some size constrained clustering problems with norm L-p. We obtain the following results: (i) A separation property for the constrained 2-clustering problem. This implies that the optimal solutions in the 1-dimensional case verify the so-called "String Property"; (ii) The NP-hardness of the constrained 2-clustering problem for every norm L-p (p > 1); (iii) A polynomial time algorithm for the constrained 2-clustering problem in dimension 1 for every norm L-p with integer p. We also give evidence that this result cannot be extended to norm L-p with rational non-integer p; (iv) The NP-hardness of the constrained clustering problem in dimension 1 for every norm L-p (p >= 1).
clustering np-hardness size constraints