Selecting a subset of diverse points based on the squared euclidean distance

In this paper we consider two closely related problems of selecting a diverse subset of points with respect to squared Euclidean distance. Given a set of points in Euclidean space, the first problem is to find a subset of a specified size M maximizing the sum of squared Euclidean distances between t...

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Vydané v:	Annals of mathematics and artificial intelligence Ročník 90; číslo 7-9; s. 965 - 977
Hlavní autori:	Eremeev, Anton V., Kel’manov, Alexander V., Kovalyov, Mikhail Y., Pyatkin, Artem V.
Médium:	Journal Article
Jazyk:	English
Vydavateľské údaje:	Cham Springer International Publishing 01.09.2022 Springer Springer Nature B.V
Predmet:	Algorithms Approximation Artificial Intelligence Complex Systems Complexity Computer Science Dynamic programming Euclidean geometry Euclidean space Hardness Integer programming Mathematics Optimization Polynomials S702: Lion14 Strong NP-hardness Given size Maximum variance Pseudo-polynomial time 62H30 Euclidean space Integer instance Fixed space dimension 68W25 Subset of points Exact algorithm 90C09
ISSN:	1012-2443, 1573-7470
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Abstract	In this paper we consider two closely related problems of selecting a diverse subset of points with respect to squared Euclidean distance. Given a set of points in Euclidean space, the first problem is to find a subset of a specified size M maximizing the sum of squared Euclidean distances between the chosen points. The second problem asks for a minimum cardinality subset of points, given a constraint on the sum of squared Euclidean distances between them. We consider the computational complexity of both problems and propose exact dynamic programming algorithms in the case of integer input data. If the dimension of the Euclidean space is bounded by a constant, these algorithms have a pseudo-polynomial time complexity. We also develop an FPTAS for the special case of the first problem, where the dimension of the Euclidean space is bounded by a constant.
AbstractList	In this paper we consider two closely related problems of selecting a diverse subset of points with respect to squared Euclidean distance. Given a set of points in Euclidean space, the first problem is to find a subset of a specified size M maximizing the sum of squared Euclidean distances between the chosen points. The second problem asks for a minimum cardinality subset of points, given a constraint on the sum of squared Euclidean distances between them. We consider the computational complexity of both problems and propose exact dynamic programming algorithms in the case of integer input data. If the dimension of the Euclidean space is bounded by a constant, these algorithms have a pseudo-polynomial time complexity. We also develop an FPTAS for the special case of the first problem, where the dimension of the Euclidean space is bounded by a constant. In this paper we consider two closely related problems of selecting a diverse subset of points with respect to squared Euclidean distance. Given a set of points in Euclidean space, the first problem is to find a subset of a specified size M maximizing the sum of squared Euclidean distances between the chosen points. The second problem asks for a minimum cardinality subset of points, given a constraint on the sum of squared Euclidean distances between them. We consider the computational complexity of both problems and propose exact dynamic programming algorithms in the case of integer input data. If the dimension of the Euclidean space is bounded by a constant, these algorithms have a pseudo-polynomial time complexity. We also develop an FPTAS for the special case of the first problem, where the dimension of the Euclidean space is bounded by a constant. Keywords Euclidean space * Subset of points * Given size * Maximum variance * Strong NP-hardness * Integer instance * Exact algorithm * Fixed space dimension * Pseudo-polynomial time Mathematics Subject Classification (2010) 62H30 * 90C09 * 68W25 In this paper we consider two closely related problems of selecting a diverse subset of points with respect to squared Euclidean distance. Given a set of points in Euclidean space, the first problem is to find a subset of a specified size M maximizing the sum of squared Euclidean distances between the chosen points. The second problem asks for a minimum cardinality subset of points, given a constraint on the sum of squared Euclidean distances between them. We consider the computational complexity of both problems and propose exact dynamic programming algorithms in the case of integer input data. If the dimension of the Euclidean space is bounded by a constant, these algorithms have a pseudo-polynomial time complexity. We also develop an FPTAS for the special case of the first problem, where the dimension of the Euclidean space is bounded by a constant.
Audience	Academic
Author	Eremeev, Anton V. Kel’manov, Alexander V. Kovalyov, Mikhail Y. Pyatkin, Artem V.
Author_xml	– sequence: 1 givenname: Anton V. orcidid: 0000-0001-5289-7874 surname: Eremeev fullname: Eremeev, Anton V. email: eremeev@ofim.oscsbras.ru organization: Sobolev Institute of Mathematics SB RAS – sequence: 2 givenname: Alexander V. surname: Kel’manov fullname: Kel’manov, Alexander V. organization: Sobolev Institute of Mathematics SB RAS – sequence: 3 givenname: Mikhail Y. surname: Kovalyov fullname: Kovalyov, Mikhail Y. organization: United Institute of Informatics Problems – sequence: 4 givenname: Artem V. surname: Pyatkin fullname: Pyatkin, Artem V. organization: Sobolev Institute of Mathematics SB RAS
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Cites_doi	10.1145/321906.321909 10.1134/S1990478914030041 10.1134/S1990478912030131 10.1134/S1990478911030069 10.2307/2528096 10.1016/0196-6774(91)90022-Q 10.1134/S0005117912020129 10.1134/S199047891604013X 10.1111/j.1540-5915.1993.tb00509.x 10.1007/s10100-009-0093-3 10.1007/978-3-030-22629-9_38 10.1007/978-3-030-53552-0_6 10.1007/978-3-030-04693-4_1
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References	CR4 Kel’manov, Romanchenko (CR12) 2012; 73 CR3 CR6 CR5 Edwards, Cavalli-Sforza (CR8) 1965; 21 Kel’manov, Pyatkin (CR11) 2011; 5 CR7 Porter, Eawal, Rachie, Wien, Willians (CR17) 1975 Kel’manov, Romanchenko (CR13) 2014; 8 Garey, Johnson (CR9) 1979 CR20 Aggarwal, Imai, Katoh, Suri (CR1) 1991; 12 Shenmaier (CR18) 2012; 6 Kuo, Glover, Dhir (CR14) 1993; 24 Shenmaier (CR19) 2016; 10 Ibarra, Kim (CR10) 1975; 22 Papadimitriou (CR16) 1994 Aringhieri (CR2) 2009; 17 McConnell (CR15) 1988; 117 AWF Edwards (9773_CR8) 1965; 21 AV Kel’manov (9773_CR13) 2014; 8 R Aringhieri (9773_CR2) 2009; 17 VV Shenmaier (9773_CR18) 2012; 6 AV Kel’manov (9773_CR12) 2012; 73 9773_CR20 9773_CR7 9773_CR6 S McConnell (9773_CR15) 1988; 117 9773_CR5 9773_CR4 9773_CR3 WM Porter (9773_CR17) 1975 VV Shenmaier (9773_CR19) 2016; 10 MR Garey (9773_CR9) 1979 O Ibarra (9773_CR10) 1975; 22 AV Kel’manov (9773_CR11) 2011; 5 CC Kuo (9773_CR14) 1993; 24 CH Papadimitriou (9773_CR16) 1994 H Aggarwal (9773_CR1) 1991; 12
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SubjectTerms	Algorithms Approximation Artificial Intelligence Complex Systems Complexity Computer Science Dynamic programming Euclidean geometry Euclidean space Hardness Integer programming Mathematics Optimization Polynomials S702: Lion14
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