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Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2019, Volume 25, Number 4, Pages 69–78
DOI: https://doi.org/10.21538/0134-4889-2019-25-4-69-78
(Mi timm1671)
 

Quadratic Euclidean 1-Mean and 1-Median 2-Clustering Problem with Constraints on the Size of the Clusters: Complexity and Approximability

A. V. Kel'manovab, A. V. Pyatkinab, V. I. Khandeevab

a Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences, Novosibirsk
b Novosibirsk State University
References:
Abstract: We consider the problem of partitioning a set of $N$ points in $d$-dimensional Euclidean space into two clusters minimizing the sum of the squared distances between each element and the center of the cluster to which it belongs. The center of the first cluster is its centroid (the geometric center). The center of the second cluster should be chosen among the points of the input set. We analyze the variant of the problem with given sizes (cardinalities) of the clusters; the sum of the sizes equals the cardinality of the input set. We prove that the problem is strongly NP-hard and there is no fully polynomial-time approximation scheme for it.
Keywords: Euclidean space, clustering, 2-partition, quadratic variation, center, centroid, median, strong NP-hardness, nonexistence of FPTAS, approximation-preserving reduction.
Funding agency Grant number
Russian Foundation for Basic Research 19-01-00308
18-31-00398
Siberian Branch of Russian Academy of Sciences 0314-2019-0014
0314-2019-0015
Ministry of Education and Science of the Russian Federation 5-100
This work was supported by the Russian Foundation for Basic Research (project nos. 19-01-00308 and 18-31-00398), by Program I.5.1 for Fundamental Research of the Siberian Branch of the Russian Academy of Sciences (project nos. 0314-2019-0014 and 0314-2019-0015), and by the Ministry of Education and Science of the Russian Federation within the Russian Academic Excellence Project.
Received: 12.08.2019
Revised: 10.09.2019
Accepted: 16.09.2019
English version:
Proceedings of the Steklov Institute of Mathematics (Supplementary issues), 2021, Volume 313, Issue 1, Pages S117–S124
DOI: https://doi.org/10.1134/S0081543821030123
Bibliographic databases:
Document Type: Article
UDC: 519.16+519.85
MSC: 68W25, 68Q25
Language: Russian
Citation: A. V. Kel'manov, A. V. Pyatkin, V. I. Khandeev, “Quadratic Euclidean 1-Mean and 1-Median 2-Clustering Problem with Constraints on the Size of the Clusters: Complexity and Approximability”, Trudy Inst. Mat. i Mekh. UrO RAN, 25, no. 4, 2019, 69–78; Proc. Steklov Inst. Math. (Suppl.), 313, suppl. 1 (2021), S117–S124
Citation in format AMSBIB
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\by A.~V.~Kel'manov, A.~V.~Pyatkin, V.~I.~Khandeev
\paper Quadratic Euclidean 1-Mean and 1-Median 2-Clustering Problem with Constraints on the Size of the Clusters: Complexity and Approximability
\serial Trudy Inst. Mat. i Mekh. UrO RAN
\yr 2019
\vol 25
\issue 4
\pages 69--78
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\crossref{https://doi.org/10.21538/0134-4889-2019-25-4-69-78}
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\transl
\jour Proc. Steklov Inst. Math. (Suppl.)
\yr 2021
\vol 313
\issue , suppl. 1
\pages S117--S124
\crossref{https://doi.org/10.1134/S0081543821030123}
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