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).

Abstract. We study the computational complexity of the problem of computing an optimal clustering {A1, A2, ..., A k } of a set of points assuming that every cluster size |Ai| belongs to a given set M of positive integers. We present a polynomial time algorithm for solving the problem in dimension 1, i.e. when the points are simply rational values, for an arbitrary set M of size constraints, which extends to the 1-norm an analogous procedure known for the 2-norm. Moreover, we prove that in the Euclidean plane, i.e. assuming dimension 2 and 2-norm, the problem is NP-hard even with size constraints set reduced to M = {2, 3}.

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We study the computational complexity of the problem of computing an optimal clustering {A 1 , A 2 , ..., A k } of a set of points assuming that every cluster size |A i | belongs to a given set M of positive integers. We present a polynomial time algorithm for solving the problem in dimension 1, i.e. when the points are simply rational values, for an arbitrary set M of size constraints, which extends to the 1 -norm an analogous procedure known for the Euclidean norm. Moreover, we prove that in dimension 2, assuming Euclidean norm, the problem is (strongly) NP-hard with size constraints M = {2, 4}. This result is extended also to the size constraints M = {2, 3} both in the case of Euclidean and 1 -norm.