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NP-Hardness of Some Euclidean Problems of Partitioning a Finite Set of Points

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Abstract

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).

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Authors and Affiliations

  1. Sobolev Institute of Mathematics, Siberian Branch, Russian Academy of Sciences, Novosibirsk, 630090, Russia

    A. V. Kel’manov & A. V. Pyatkin

  2. Novosibirsk State University, Novosibirsk, 630090, Russia

    A. V. Kel’manov & A. V. Pyatkin

Authors
  1. A. V. Kel’manov
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  2. A. V. Pyatkin
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Correspondence to A. V. Kel’manov.

Additional information

Original Russian Text © A.V. Kel’manov, A.V. Pyatkin, 2018, published in Zhurnal Vychislitel’noi Matematiki i Matematicheskoi Fiziki, 2018, Vol. 58, No. 5, pp. 852–856.

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Kel’manov, A.V., Pyatkin, A.V. NP-Hardness of Some Euclidean Problems of Partitioning a Finite Set of Points. Comput. Math. and Math. Phys. 58, 822–826 (2018). https://doi.org/10.1134/S0965542518050123

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  • DOI: https://doi.org/10.1134/S0965542518050123

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