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Zh. Vychisl. Mat. Mat. Fiz., 2016, Volume 56, Number 2, Pages 332–340 (Mi zvmmf10348)  

This article is cited in 11 scientific papers (total in 11 papers)

Fully polynomial-time approximation scheme for a special case of a quadratic Euclidean 2-clustering problem

A. V. Kel'manovab, V. I. Khandeevab

a Sobolev Institute of Mathematics, Siberian Branch, Russian Academy of Sciences, pr. Akademika Koptyuga 4, Novosibirsk, 630090, Russia
b Novosibirsk State University, ul. Pirogova 2, Novosibirsk, 630090, Russia

Abstract: The strongly NP-hard problem of partitioning a finite set of points of Euclidean space into two clusters of given sizes (cardinalities) minimizing the sum (over both clusters) of the intracluster sums of squared distances from the elements of the clusters to their centers is considered. It is assumed that the center of one of the sought clusters is specified at the desired (arbitrary) point of space (without loss of generality, at the origin), while the center of the other one is unknown and determined as the mean value over all elements of this cluster. It is shown that unless P = NP, there is no fully polynomial-time approximation scheme for this problem, and such a scheme is substantiated in the case of a fixed space dimension.

Key words: cluster analysis, partition, Euclidean space, minimum of the sum of squares of distances, NP-hardness, fixed space dimension, FPTAS.

Funding Agency Grant Number
Russian Foundation for Basic Research 13-07-00070_а
15-01-00462_а


DOI: https://doi.org/10.7868/S0044466916020113

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English version:
Computational Mathematics and Mathematical Physics, 2016, 56:2, 334–341

Bibliographic databases:

UDC: 519.7
Received: 02.03.2015

Citation: A. V. Kel'manov, V. I. Khandeev, “Fully polynomial-time approximation scheme for a special case of a quadratic Euclidean 2-clustering problem”, Zh. Vychisl. Mat. Mat. Fiz., 56:2 (2016), 332–340; Comput. Math. Math. Phys., 56:2 (2016), 334–341

Citation in format AMSBIB
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. A. V. Kel'manov, S. A. Khamidullin, V. I. Khandeev, “Fully polynomial-time approximation scheme for a sequence $2$-clustering problem”, J. Appl. Industr. Math., 10:2 (2016), 209–219  mathnet  crossref  crossref  mathscinet  elib
    2. A. Kel'manov, A. Motkova, “A fully polynomial-time approximation scheme for a special case of a balanced 2-clustering problem”, Discrete Optimization and Operations Research, Lecture Notes in Computer Science, 9869, ed. Y. Kochetov, M. Khachay, V. Beresnev, E. Nurminski, P. Pardalos, Springer Int Publishing Ag, 2016, 182–192  crossref  mathscinet  zmath  isi  scopus
    3. A. V. Kel'manov, A. V. Motkova, V. V. Shenmaier, “Approximation scheme for the problem of weighted 2-partitioning with a fixed center of one cluster”, Proc. Steklov Inst. Math. (Suppl.), 303, suppl. 1 (2018), 136–145  mathnet  crossref  crossref  isi  elib
    4. A. V. Eremeev, A. V. Kel'manov, A. V. Pyatkin, “On complexity of searching a subset of vectors with shortest average under a cardinality restriction”, Analysis of Images, Social Networks and Texts, AIST 2016, Communications in Computer and Information Science, 661, eds. D. Ignatov, M. Khachay, V. Labunets, N. Loukachevitch, S. Nikolenko, A. Panchenko, A. Savchenko, et, Springler, 2017, 51–57  crossref  isi
    5. A. Kel'manov, “Efficient approximation algorithms for some NP-hard problems of partitioning a set and a sequence”, 2017 International Multi-Conference on Engineering, Computer and Information Sciences, SIBIRCON 2017, IEEE, 2017, 87–90  crossref  isi
    6. A. Kel'manov, V. Khandeev, “Some algorithms with guaranteed accuracy for 2-clustering problems with given center of one cluster”, 2017 International Multi-Conference on Engineering, Computer and Information Sciences, SIBIRCON 2017, IEEE, 2017, 91–93  crossref  isi
    7. A. Kel'manov, A. Motkova, “An approximation polynomial-time algorithm for a cardinality-weighted 2-clustering problem”, 2017 International Multi-Conference on Engineering, Computer and Information Sciences, SIBIRCON 2017, IEEE, 2017, 94–96  crossref  isi
    8. A. Kel'manov, S. Khamidullin, V. Khandeev, “A randomized algorithm for 2-partition of a sequence”, Analysis of Images, Social Networks and Texts, AIST 2017, Lecture Notes in Computer Science, 10716, eds. W. VanDerAalst, D. Ignatov, M. Khachay, S. Kuznetsov, V. Lempitsky, I. Lomazova, N. Loukachevitch, Springer, 2018, 313–322  crossref  isi  scopus
    9. A. Kel'manov, A. Motkova, V. Shenmaier, “An approximation scheme for a weighted two-cluster partition problem”, Analysis of Images, Social Networks and Texts, AIST 2017, Lecture Notes in Computer Science, 10716, eds. W. VanDerAalst, D. Ignatov, M. Khachay, S. Kuznetsov, V. Lempitsky, I. Lomazova, N. Loukachevitch, A. Napoli, A. Panchenko, P. Pardalos, A. Savchenko, S. Wasserman, Springer, 2018, 323–333  crossref  isi  scopus
    10. A. V. Kel'manov, A. V. Motkova, “Polynomial-time approximation algorithm for the problem of cardinality-weighted variance-based 2-clustering with a given center”, Comput. Math. Math. Phys., 58:1 (2018), 130–136  mathnet  crossref  crossref  isi  elib
    11. A. V. Kel'manov, S. A. Khamidullin, V. I. Khandeev, “A randomized algorithm for a sequence 2-clustering problem”, Comput. Math. Math. Phys., 58:12 (2018), 2078–2085  mathnet  crossref  crossref  isi  elib
  • Журнал вычислительной математики и математической физики Computational Mathematics and Mathematical Physics
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