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

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 assume...

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Bibliographic Details
Published in:Computational mathematics and mathematical physics Vol. 56; no. 2; pp. 334 - 341
Main Authors: Kel’manov, A. V., Khandeev, V. I.
Format: Journal Article
Language:English
Published: Moscow Pleiades Publishing 01.02.2016
Springer Nature B.V
Subjects:
Algorithms
Approximation
Cluster analysis
Clustering
Clusters
Computation
Computational mathematics
Computational Mathematics and Numerical Analysis
Data analysis
Euclidean geometry
Euclidean space
Hypotheses
Hypothesis testing
Mathematical analysis
Mathematical models
Mathematics
Mathematics and Statistics
Origins
Partitioning
Studies
NP-hardness
minimum of the sum of squares of distances
partition
Euclidean space
fixed space dimension
cluster analysis
FPTAS
ISSN:0965-5425, 1555-6662
Online Access:Get full text
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Summary: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.
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ISSN:0965-5425
1555-6662
DOI:10.1134/S0965542516020111