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FPT-algorithms for some problems related to integer programming

Author

Listed:
  • D. V. Gribanov

    (Lobachevsky State University of Nizhny Novgorod
    National Research University Higher School of Economics)

  • D. S. Malyshev

    (National Research University Higher School of Economics)

  • P. M. Pardalos

    (National Research University Higher School of Economics
    University of Florida)

  • S. I. Veselov

    (Lobachevsky State University of Nizhny Novgorod)

Abstract

In this paper, we present fixed-parameter tractable algorithms for special cases of the shortest lattice vector, integer linear programming, and simplex width computation problems, when matrices included in the problems’ formulations are near square. The parameter is the maximum absolute value of the rank minors in the corresponding matrices. Additionally, we present fixed-parameter tractable algorithms with respect to the same parameter for the problems, when the matrices have no singular rank submatrices.

Suggested Citation

  • D. V. Gribanov & D. S. Malyshev & P. M. Pardalos & S. I. Veselov, 2018. "FPT-algorithms for some problems related to integer programming," Journal of Combinatorial Optimization, Springer, vol. 35(4), pages 1128-1146, May.
    • Handle: RePEc:spr:jcomop:v:35:y:2018:i:4:d:10.1007_s10878-018-0264-z
    DOI: 10.1007/s10878-018-0264-z
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    References listed on IDEAS

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    1. Ravi Kannan, 1987. "Minkowski's Convex Body Theorem and Integer Programming," Mathematics of Operations Research, INFORMS, vol. 12(3), pages 415-440, August.
    2. H. W. Lenstra, 1983. "Integer Programming with a Fixed Number of Variables," Mathematics of Operations Research, INFORMS, vol. 8(4), pages 538-548, November.
    3. Éva Tardos, 1986. "A Strongly Polynomial Algorithm to Solve Combinatorial Linear Programs," Operations Research, INFORMS, vol. 34(2), pages 250-256, April.
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    Cited by:

    1. Dmitry Gribanov & Dmitry Malyshev & Ivan Shumilov, 2024. "On a Simple Connection Between $$\Delta$$ Δ -Modular ILP and LP, and a New Bound on the Number of Integer Vertices," SN Operations Research Forum, Springer, vol. 5(2), pages 1-9, June.

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