Problems of partitioning a finite set of Euclidean points (vectors) into clusters are considered. The criterion is to minimize the sum, over all clusters, of (1) squared norms of the sums of cluster elements normalized by the cardinality, (2) squared norms of the sums of cluster elements, and (3) norms of the sum of cluster elements. It is proved that all these problems are strongly NP-hard if the number of clusters is a part of the input and are NP-hard in the ordinary sense if the number of clusters is not a part of the input (is fixed). Moreover, the problems are NP-hard even in the case of dimension 1 (on a line).
partitioning Euclidean space norm of sum NP-hardness
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A. V. Kel’manov and A. V. Pyatkin, “Complexity of certain problems of searching for subsets of vectors and cluster analysis,” Comput. Math. Math. Phys. 49 (11), 1966–1971 (2009).MathSciNetCrossRefzbMATHGoogle Scholar