Polynomial-Time Approximation Algorithm for the Problem of Cardinality-Weighted Variance-Based 2-Clustering with a Given Center

A strongly NP-hard problem of partitioning a finite set of points of Euclidean space into two clusters is considered. The solution criterion is the minimum of the sum (over both clusters) of weighted sums of squared distances from the elements of each cluster to its geometric center. The weights of...

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Bibliographic Details
Published in:Computational mathematics and mathematical physics Vol. 58; no. 1; pp. 130 - 136
Main Authors: Kel’manov, A. V., Motkova, A. V.
Format: Journal Article
Language:English
Published: Moscow Pleiades Publishing 01.01.2018
Springer Nature B.V
Subjects:
Approximation
Clustering
Computational Mathematics and Numerical Analysis
Euclidean geometry
Euclidean space
Mathematical analysis
Mathematics
Mathematics and Statistics
Polynomials
Ratios
Sums
NP-hardness
Euclidean space
weighted 2-clustering
polynomial-time 2-approximation algorithm
ISSN:0965-5425, 1555-6662
Online Access:Get full text
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Summary:A strongly NP-hard problem of partitioning a finite set of points of Euclidean space into two clusters is considered. The solution criterion is the minimum of the sum (over both clusters) of weighted sums of squared distances from the elements of each cluster to its geometric center. The weights of the sums are equal to the cardinalities of the desired clusters. The center of one cluster is given as input, while the center of the other is unknown and is determined as the point of space equal to the mean of the cluster elements. A version of the problem is analyzed in which the cardinalities of the clusters are given as input. A polynomial-time 2-approximation algorithm for solving the problem is constructed.
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ISSN:0965-5425
1555-6662
DOI:10.1134/S0965542518010074