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 \(\ell _1\)-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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- 1.
If X does not admit a \(\mathcal {M}\)-clustering then symbol \(\bot \) is returned.
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Acknowledgments
We thank an anonymous referee for his/her useful comments on other problems related to clustering with relaxed size constraints.
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Goldwurm, M., Lin, J., Saccà, F. (2016). On the Complexity of Clustering with Relaxed Size Constraints. In: Dondi, R., Fertin, G., Mauri, G. (eds) Algorithmic Aspects in Information and Management. AAIM 2016. Lecture Notes in Computer Science(), vol 9778. Springer, Cham. https://doi.org/10.1007/978-3-319-41168-2_3
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DOI: https://doi.org/10.1007/978-3-319-41168-2_3
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