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 Diskretn. Anal. Issled. Oper., 2016, Volume 23, Number 2, Pages 21–40 (Mi da843)

Fully polynomial-time approximation scheme for a sequence $2$-clustering problem

A. V. Kel'manovab, S. A. Khamidullinb, V. I. Khandeevb

a Novosibirsk State University, 2 Pirogova St., 630090 Novosibirsk, Russia
b Sobolev Institute of Mathematics, 4 Koptyug Ave., 630090 Novosibirsk, Russia

Abstract: We consider a strongly NP-hard problem of partitioning a finite sequence of points in Euclidean space into two clusters minimizing the sum over both clusters of intra-cluster sum of squared distances from the clusters elements to their centers. The sizes of the clusters are fixed. The centroid of the first cluster is defined as the mean value of all vectors in the cluster, and the center of the second one is given in advance and is equal to 0. Additionally, the partition must satisfy the restriction that for all vectors in the first cluster the difference between the indices of two consequent points from this cluster is bounded from below and above by some given constants. We present a fully polynomial-time approximation scheme for the case of fixed space dimension. Bibliogr. 27.

Keywords: partitioning, sequence, Euclidean space, minimum sum-of-squared distances, NP-hardness, FPTAS.

 Funding Agency Grant Number Russian Foundation for Basic Research 15-01-0046216-07-0016816-31-00186-ìîë-à

DOI: https://doi.org/10.17377/daio.2016.23.511

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English version:
Journal of Applied and Industrial Mathematics, 2016, 10:2, 209–219

Bibliographic databases:

Document Type: Article
UDC: 519.16+519.85
Revised: 12.01.2016

Citation: A. V. Kel'manov, S. A. Khamidullin, V. I. Khandeev, “Fully polynomial-time approximation scheme for a sequence $2$-clustering problem”, Diskretn. Anal. Issled. Oper., 23:2 (2016), 21–40; J. Appl. Industr. Math., 10:2 (2016), 209–219

Citation in format AMSBIB
\Bibitem{KelKhaKha16} \by A.~V.~Kel'manov, S.~A.~Khamidullin, V.~I.~Khandeev \paper Fully polynomial-time approximation scheme for a~sequence $2$-clustering problem \jour Diskretn. Anal. Issled. Oper. \yr 2016 \vol 23 \issue 2 \pages 21--40 \mathnet{http://mi.mathnet.ru/da843} \crossref{https://doi.org/10.17377/daio.2016.23.511} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=3557592} \elib{http://elibrary.ru/item.asp?id=26129765} \transl \jour J. Appl. Industr. Math. \yr 2016 \vol 10 \issue 2 \pages 209--219 \crossref{https://doi.org/10.1134/S199047891602006X} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84971334424} 

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This publication is cited in the following articles:
1. A. V. Kel'manov, L. V. Mikhailova, S. A. Khamidullin, V. I. Khandeev, “An approximation algorithm for the problem of partitioning a sequence into clusters with constraints on their cardinalities”, Proc. Steklov Inst. Math. (Suppl.), 299, suppl. 1 (2017), 88–96
2. 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), IEEE, 2017, 87–90
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