Abstract
We consider the strongly NP-hard problem of partitioning a finite set of Euclidean points into two clusters so as to minimize the sum (over both clusters) of the weighted sums of the squared intra-cluster distances from the elements of the cluster to its center. The weights of the sums are equal to the cardinalities of the clusters. The center of one of the clusters is given as input, while the center of the other cluster is unknown and is determined as the mean value over all points in this cluster, i.e., as the geometric center (centroid). The version of the problem with constrained cardinalities of the clusters is analyzed. We construct an approximation algorithm for the problem and show that it is a fully polynomial-time approximation scheme (FPTAS) if the space dimension is bounded by a constant.
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Alexander Vasilyevich Kel’manov. Born 1952. Graduated from Izhevsk State Technical University in 1974 with specialty in Applied Mathematics. Received Candidate’s Degree in Engineering Cybernetics and Information Theory in 1980 and Doctor of Sciences degree in Physics and Mathematics in 1994. Currently with Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences, head of Data Analysis Laboratory. Scientific interests: data analysis, data mining, pattern recognition, clusterization, discrete optimization, NP-hard problems, efficient algorithms with performance guarantees. Author of more than 200 publications.
Anna Vladimirovna Motkova. Born 1993. Graduated from Novosibirsk State University in 2017 with specialty in Mathematics. Currently with Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences, Ph.D. student in Data Analysis Laboratory. Scientific interests: data analysis, pattern recognition, clustering, discrete optimization, NP-hard problems, efficient algorithms with performance guarantees. Author of 8 publications.
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Kel’manov, A.V., Motkova, A.V. Approximation Scheme for a Quadratic Euclidean Weighted 2-Clustering Problem. Pattern Recognit. Image Anal. 28, 17–23 (2018). https://doi.org/10.1134/S105466181801008X
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DOI: https://doi.org/10.1134/S105466181801008X