On Finding Maximum Cardinality Subset of Vectors with a Constraint on Normalized Squared Length of Vectors Sum

  • Anton V. EremeevEmail author
  • Alexander V. Kelmanov
  • Artem V. Pyatkin
  • Igor A. Ziegler
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10716)


In this paper, we consider the problem of finding a maximum cardinality subset of vectors, given a constraint on the normalized squared length of vectors sum. This problem is closely related to Problem 1 from (Eremeev, Kel’manov, Pyatkin, 2016). The main difference consists in swapping the constraint with the optimization criterion.

We prove that the problem is NP-hard even in terms of finding a feasible solution. An exact algorithm for solving this problem is proposed. The algorithm has a pseudo-polynomial time complexity in the special case of the problem, where the dimension of the space is bounded from above by a constant and the input data are integer. A computational experiment is carried out, where the proposed algorithm is compared to COINBONMIN solver, applied to a mixed integer quadratic programming formulation of the problem. The results of the experiment indicate superiority of the proposed algorithm when the dimension of Euclidean space is low, while the COINBONMIN has an advantage for larger dimensions.


Vectors sum Subset selection Euclidean norm NP-hardness Pseudo-polymonial time 



This research is supported by RFBR, projects 15-01-00462, 16-01-00740 and 15-01-00976.


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Copyright information

© Springer International Publishing AG 2018

Authors and Affiliations

  • Anton V. Eremeev
    • 1
    • 2
    Email author
  • Alexander V. Kelmanov
    • 3
    • 4
  • Artem V. Pyatkin
    • 3
    • 4
  • Igor A. Ziegler
    • 1
    • 2
  1. 1.Omsk Branch of Sobolev Institute of Mathematics SB RASOmskRussia
  2. 2.Omsk State University n.a. F.M. DostoevskyOmskRussia
  3. 3.Sobolev Institute of Mathematics SB RASNovosibirskRussia
  4. 4.Novosibirsk State UniversityNovosibirskRussia

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