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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| Published in: | Computational mathematics and mathematical physics Vol. 58; no. 1; pp. 130 - 136 |
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| Main Authors: | , |
| Format: | Journal Article |
| Language: | English |
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Pleiades Publishing
01.01.2018
Springer Nature B.V |
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| ISSN: | 0965-5425, 1555-6662 |
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| Abstract | 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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| AbstractList | 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. |
| Author | Motkova, A. V. Kel’manov, A. V. |
| Author_xml | – sequence: 1 givenname: A. V. surname: Kel’manov fullname: Kel’manov, A. V. email: kelm@math.nsc.ru organization: Sobolev Institute of Mathematics, Siberian Branch, Russian Academy of Sciences, Novosibirsk State University – sequence: 2 givenname: A. V. surname: Motkova fullname: Motkova, A. V. organization: Sobolev Institute of Mathematics, Siberian Branch, Russian Academy of Sciences, Novosibirsk State University |
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| Cites_doi | 10.1007/978-3-642-95322-4_5 10.1134/S096554251502013X 10.1134/S0081543816090066 10.1134/S0965542516020111 10.1134/S1990478915040067 10.1080/01621459.1971.10482319 10.1134/S1990478916030054 10.1134/S1990478913040066 10.1134/S096554251603009X 10.1134/S1990478911040107 10.1134/S1990478914030041 10.1134/S1064562415050233 10.1007/978-3-319-14142-8 10.1007/978-3-319-44914-2_15 10.1145/321958.321975 10.1007/s10994-009-5103-0 10.1007/978-0-387-21606-5 |
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| Copyright | Pleiades Publishing, Ltd. 2018 Computational Mathematics and Mathematical Physics is a copyright of Springer, (2018). All Rights Reserved. |
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| SubjectTerms | Approximation Clustering Computational Mathematics and Numerical Analysis Euclidean geometry Euclidean space Mathematical analysis Mathematics Mathematics and Statistics Polynomials Ratios Sums |
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| Title | Polynomial-Time Approximation Algorithm for the Problem of Cardinality-Weighted Variance-Based 2-Clustering with a Given Center |
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