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 algorithm for the constrained 2-clustering problem in 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).
Size constrained distance clustering : separation properties and some complexity results / A. Bertoni, M. Goldwurm, J. Lin, F. Saccà. - In: FUNDAMENTA INFORMATICAE. - ISSN 0169-2968. - 115:1(2012), pp. 125-139. [10.3233/FI-2012-644]
Size constrained distance clustering : separation properties and some complexity results
A. BertoniPrimo
;M. GoldwurmSecondo
;J. LinPenultimo
;F. SaccàUltimo
2012
Abstract
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 algorithm for the constrained 2-clustering problem in 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).File | Dimensione | Formato | |
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