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NP-Hardness of the Problem of Optimal Box Positioning

Author

Listed:
  • Alexei V. Galatenko

    (Faculty of Mechanics and Mathematics, Lomonosov Moscow State University, Leninskie Gory 1, 119991 Moscow, Russia)

  • Stepan A. Nersisyan

    (Faculty of Mechanics and Mathematics, Lomonosov Moscow State University, Leninskie Gory 1, 119991 Moscow, Russia)

  • Dmitriy N. Zhuk

    (Faculty of Mechanics and Mathematics, Lomonosov Moscow State University, Leninskie Gory 1, 119991 Moscow, Russia)

Abstract

We consider the problem of finding a position of a d -dimensional box with given edge lengths that maximizes the number of enclosed points of the given finite set P ⊂ R d , i.e., the problem of optimal box positioning. We prove that while this problem is polynomial for fixed values of d , it is NP-hard in the general case. The proof is based on a polynomial reduction technique applied to the considered problem and the 3-CNF satisfiability problem.

Suggested Citation

  • Alexei V. Galatenko & Stepan A. Nersisyan & Dmitriy N. Zhuk, 2019. "NP-Hardness of the Problem of Optimal Box Positioning," Mathematics, MDPI, vol. 7(8), pages 1-6, August.
    • Handle: RePEc:gam:jmathe:v:7:y:2019:i:8:p:711-:d:255294
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    References listed on IDEAS

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    1. Stepan A. Nersisyan & Vera V. Pankratieva & Vladimir M. Staroverov & Vladimir E. Podolskii, 2017. "A Greedy Clustering Algorithm Based on Interval Pattern Concepts and the Problem of Optimal Box Positioning," Journal of Applied Mathematics, Hindawi, vol. 2017, pages 1-9, September.
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