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Discrete Optimization: The Case of Generalized BCC Lattice

Author

Listed:
  • Gergely Kovács

    (Department of Methodology of Applied Sciences, Edutus University, 2800 Tatabánya, Hungary
    These authors contributed equally to this work.)

  • Benedek Nagy

    (Department of Mathematics, Faculty of Arts and Sciences, Eastern Mediterranean University, Famagusta 99628, North Cyprus, Turkey
    These authors contributed equally to this work.)

  • Gergely Stomfai

    (ELTE Apáczai Csere János High School, 1053 Budapest, Hungary
    These authors contributed equally to this work.)

  • Neşet Deniz Turgay

    (Department of Mathematics, Faculty of Arts and Sciences, Eastern Mediterranean University, Famagusta 99628, North Cyprus, Turkey
    These authors contributed equally to this work.)

  • Béla Vizvári

    (Department of Industrial Engineering, Eastern Mediterranean University, Famagusta 99628, North Cyprus, Turkey
    These authors contributed equally to this work.)

Abstract

Recently, operations research, especially linear integer-programming, is used in various grids to find optimal paths and, based on that, digital distance. The 4 and higher-dimensional body-centered-cubic grids is the n D ( n ≥ 4 ) equivalent of the 3D body-centered cubic grid, a well-known grid from solid state physics. These grids consist of integer points such that the parity of all coordinates are the same: either all coordinates are odd or even. A popular type digital distance, the chamfer distance, is used which is based on chamfer paths. There are two types of neighbors (closest same parity and closest different parity point-pairs), and the two weights for the steps between the neighbors are fixed. Finding the minimal path between two points is equivalent to an integer-programming problem. First, we solve its linear programming relaxation. The optimal path is found if this solution is integer-valued. Otherwise, the Gomory-cut is applied to obtain the integer-programming optimum. Using the special properties of the optimization problem, an optimal solution is determined for all cases of positive weights. The geometry of the paths are described by the Hilbert basis of the non-negative part of the kernel space of matrix of steps.

Suggested Citation

  • Gergely Kovács & Benedek Nagy & Gergely Stomfai & Neşet Deniz Turgay & Béla Vizvári, 2021. "Discrete Optimization: The Case of Generalized BCC Lattice," Mathematics, MDPI, vol. 9(3), pages 1-20, January.
    • Handle: RePEc:gam:jmathe:v:9:y:2021:i:3:p:208-:d:483815
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    References listed on IDEAS

    as
    1. Robert G. Jeroslow, 1978. "Some Basis Theorems for Integral Monoids," Mathematics of Operations Research, INFORMS, vol. 3(2), pages 145-154, May.
    2. Gergely Kovács & Benedek Nagy & Béla Vizvári, 2019. "Chamfer distances on the isometric grid: a structural description of minimal distances based on linear programming approach," Journal of Combinatorial Optimization, Springer, vol. 38(3), pages 867-886, October.
    Full references (including those not matched with items on IDEAS)

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