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

• Published in: Annals of mathematics and artificial intelligence Vol. 90; no. 7-9; pp. 965 - 977
• Main Authors: Eremeev, Anton V., Kel’manov, Alexander V., Kovalyov, Mikhail Y., Pyatkin, Artem V.
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 01.09.2022; Springer; Springer Nature B.V
• Subjects: 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
• DOI: 10.1007/s10472-021-09773-z

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.