On the complexity of clustering with relaxed size constraints in fixed dimension

We study the computational complexity of the problem of computing an optimal clustering {A1,A2,...,Ak} 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 ℓ1-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 ℓ1-norm.