On the complexity of clustering with relaxed size constraints

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 ℓ2-norm. Moreover, we prove that in the Euclidean plane, i.e. assuming dimension 2 and ℓ2-norm, the problem is NP-hard even with size constraints set reduced to M = {2, 3}.