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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)

Abstract

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.

Keywords

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

Notes

Acknowledgements

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

References

  1. 1.
    Ageev, A.A., Kel’manov, A.V., Pyatkin, A.V., Khamidullin, S.A., Shenmaier, V.V.: Polynomial approximation algorithm for the data editing and data cleaning problem. Pattern Recogn. Image Anal. 27(3), 365–370 (2017)CrossRefGoogle Scholar
  2. 2.
    Borisovsky, P.A., Eremeev, A.V., Grinkevich, E.B., Klokov, S.A., Vinnikov, A.V.: Trading hubs construction for electricity markets. In: Kallrath, J., Pardalos, P.M., Rebennack, S., Scheidt, M. (eds.) Optimization in the Energy Industry, pp. 29–58. Springer, Heidelberg (2009).  https://doi.org/10.1007/978-3-540-88965-6_3 CrossRefGoogle Scholar
  3. 3.
    Borisovsky, P.A., Eremeev, A.V., Grinkevich, E.B., Klokov, S.A., Kosarev, N.A.: Trading hubs construction in electricity markets using evolutionary algorithms. Pattern Recogn. Image Anal. 24(2), 270–282 (2014)CrossRefGoogle Scholar
  4. 4.
    Eremeev, A.V., Kel’manov, A.V., Pyatkin, A.V.: On complexity of searching a subset of vectors with shortest average under a cardinality restriction. In: Ignatov, D.I., Khachay, M.Y., Labunets, V.G., Loukachevitch, N., Nikolenko, S.I., Panchenko, A., Savchenko, A.V., Vorontsov, K. (eds.) AIST 2016. CCIS, vol. 661, pp. 51–57. Springer, Cham (2017).  https://doi.org/10.1007/978-3-319-52920-2_5 CrossRefGoogle Scholar
  5. 5.
    Eremeev, A.V., Kel’manov, A.V., Pyatkin, A.V.: On the complexity of some Euclidean optimal summing problems. Dokl. Math. 93(3), 286–288 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman and Company, San Francisco (1979)zbMATHGoogle Scholar
  7. 7.
    Greco, L.: Robust Methods for Data Reduction. Chapman and Hall/CRC, Boca Raton (2015)zbMATHGoogle Scholar
  8. 8.
    NEPOOL Energy Market Hub White Paper and Proposal. Hub Analysis Working Group NEPOOL Markets Committee (2003)Google Scholar
  9. 9.
    Osborne, J.W.: Best Practices in Data Cleaning: A Complete Guide to Everything You Need to Do Before and After Collecting Your Data, 1st edn. SAGE Publication, Inc., Los Angeles (2013)CrossRefGoogle Scholar
  10. 10.
    de Waal, T., Pannekoek, J., Scholtus, S.: Handbook of Statistical Data Editing and Imputation. John Wiley and Sons, Inc., Hoboken (2011)CrossRefGoogle Scholar

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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