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

Abstract

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.