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

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Analysis of Images, Social Networks and Texts (AIST 2017)

Part of the book series: Lecture Notes in Computer Science ((LNISA,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.

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References

  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)

    Article  Google Scholar 

  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

    Chapter  Google Scholar 

  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)

    Article  Google Scholar 

  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

    Chapter  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    MATH  Google Scholar 

  7. Greco, L.: Robust Methods for Data Reduction. Chapman and Hall/CRC, Boca Raton (2015)

    MATH  Google Scholar 

  8. NEPOOL Energy Market Hub White Paper and Proposal. Hub Analysis Working Group NEPOOL Markets Committee (2003)

    Google Scholar 

  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)

    Book  Google Scholar 

  10. de Waal, T., Pannekoek, J., Scholtus, S.: Handbook of Statistical Data Editing and Imputation. John Wiley and Sons, Inc., Hoboken (2011)

    Book  Google Scholar 

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Acknowledgements

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

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Authors and Affiliations

  1. Omsk Branch of Sobolev Institute of Mathematics SB RAS, Omsk, Russia

    Anton V. Eremeev & Igor A. Ziegler

  2. Omsk State University n.a. F.M. Dostoevsky, Omsk, Russia

    Anton V. Eremeev & Igor A. Ziegler

  3. Sobolev Institute of Mathematics SB RAS, Novosibirsk, Russia

    Alexander V. Kelmanov & Artem V. Pyatkin

  4. Novosibirsk State University, Novosibirsk, Russia

    Alexander V. Kelmanov & Artem V. Pyatkin

Authors
  1. Anton V. Eremeev
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  2. Alexander V. Kelmanov
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  3. Artem V. Pyatkin
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  4. Igor A. Ziegler
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Corresponding author

Correspondence to Anton V. Eremeev .

Editor information

Editors and Affiliations

  1. Eindhoven University of Technology, Eindhoven, The Netherlands

    Wil M.P. van der Aalst

  2. National Research University Higher School of Economics, Moscow, Russia

    Dmitry I. Ignatov

  3. Krasovsky Institute of Mathematics and Mechanics, Ekaterinburg, Russia

    Michael Khachay

  4. National Research University Higher School of Economics, Moscow, Russia

    Sergei O. Kuznetsov

  5. Skolkovo Institute of Science and Technology, Moscow, Russia

    Victor Lempitsky

  6. National Research University Higher School of Economics, Moscow, Russia

    Irina A. Lomazova

  7. Moscow State University, Moscow, Russia

    Natalia Loukachevitch

  8. LORIA, Campus Scientifique, Vandœuvre lès Nancy, France

    Amedeo Napoli

  9. University of Hamburg, Hamburg, Germany

    Alexander Panchenko

  10. University of Florida, Gainesville, Florida, USA

    Panos M. Pardalos

  11. National Research University Higher School of Economics, Nizhny Novgorod, Russia

    Andrey V. Savchenko

  12. Indiana University, Bloomington, Indiana, USA

    Stanley Wasserman

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Eremeev, A.V., Kelmanov, A.V., Pyatkin, A.V., Ziegler, I.A. (2018). On Finding Maximum Cardinality Subset of Vectors with a Constraint on Normalized Squared Length of Vectors Sum. In: van der Aalst, W., et al. Analysis of Images, Social Networks and Texts. AIST 2017. Lecture Notes in Computer Science(), vol 10716. Springer, Cham. https://doi.org/10.1007/978-3-319-73013-4_13

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  • DOI: https://doi.org/10.1007/978-3-319-73013-4_13

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-73012-7

  • Online ISBN: 978-3-319-73013-4

  • eBook Packages: Computer ScienceComputer Science (R0)

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