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Size Constrained Distance Clustering: Separation Properties and Some Complexity Results

Bertoni A., Goldwurm M., Lin J., Sacca F.

Fundamenta Informaticae

2012

Vol. 115, nr 1

125-139

In this paper we study the complexity of some size constrained clustering problems with norm Lp. 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 Lp (p > 1); (iii) A polynomial time algorithmfor the constrained 2-clustering problemin dimension 1 for every norm Lp with integer p. We also give evidence that this result cannot be extended to norm Lp with rational non-integer p; (iv) The NP-hardness of the constrained clustering problem in dimension 1 for every norm Lp (p ≥1).