NP-Completeness of Some Problems of Partitioning a Finite Set of Points in Euclidean Space into Balanced Clusters
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We consider three related problems of partitioning an \(N\)-element set of points in \(d\)-dimensional Euclidean space into two clusters balancing the value of (1) the intracluster quadratic variance normalized by the cluster size in the first problem; (2) the intracluster quadratic variance in the second problem; and (3) the size-weighted intracluster quadratic variance in the third problem. The NP-completeness of all these problems is proved.
This work was supported by the Russian Foundation for Basic Research (project nos. 19-01-00308 and 18-31-00398), by Basic Research Programs of the Russian Academy of Sciences (project nos. 0314-2019-0015 and 0314-2019-0014), and by the Top-5-100 Program of the Ministry of Education and Science of the Russian Federation.
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