On the Complexity of Clustering with Relaxed Size Constraints

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 \(\ell _1\)-norm an analogous procedure known for the \(\ell _2\)-norm. Moreover, we prove that in the Euclidean plane, i.e. assuming dimension 2 and \(\ell _2\)-norm, the problem is NP-hard even with size constraints set reduced to \(M=\{2,3\}\).

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