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Shortest Integer Vectors

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Abstract

Let A be a fixed integer matrix of size m by n and consider all b for which the body is full dimensional. We examine the set of shortest non-zero integral vectors with respect to the family of norms. We show that the number of such shortest vectors is polynomial in the bit size of A, for fixed n. We also show the existence, for any n, of a family of matrices M for which the number of shortest vectors has as a lower bound a polynomial in the bit size of M of the same degree at the polynomial bound.

Suggested Citation

  • Herbert E. Scarf & Shallcross, David F., 1991. "Shortest Integer Vectors," Cowles Foundation Discussion Papers 965, Cowles Foundation for Research in Economics, Yale University.
    • Handle: RePEc:cwl:cwldpp:965
    Note: CFP 848.
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    File URL: https://cowles.yale.edu/sites/default/files/files/pub/d09/d0965.pdf
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    References listed on IDEAS

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    1. Imre Barany & Roger Howe & Laszlo Lovasz, 1989. "On Integer Points in Polyhedra: A Lower Bound," Cowles Foundation Discussion Papers 917, Cowles Foundation for Research in Economics, Yale University.
    2. Herbert E. Scarf & R. Kannan & Laszlo Lovasz, 1988. "The Shapes of Polyhedra," Cowles Foundation Discussion Papers 883, Cowles Foundation for Research in Economics, Yale University.
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    More about this item

    Keywords

    Indivisibilities; integer programming; geometry; numbers;
    All these keywords.

    JEL classification:

    • C60 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - General
    • C61 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Optimization Techniques; Programming Models; Dynamic Analysis

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