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On a Simple Connection Between $$\Delta$$ Δ -Modular ILP and LP, and a New Bound on the Number of Integer Vertices

Author

Listed:
  • Dmitry Gribanov

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

  • Dmitry Malyshev

    (National Research University Higher School of Economics
    Huawei)

  • Ivan Shumilov

    (Lobachevsky State University of Nizhny Novgorod)

Abstract

In our note, we present a very simple and short proof of a new interesting fact about the faces of an integer hull of a given rational polyhedron. This fact has a complete analog in linear programming theory and can be useful to establish new constructive upper bounds on the number of vertices in an integer hull of a $$\Delta$$ Δ -modular polyhedron, which are competitive for small values of $$\Delta$$ Δ and can be useful for integer linear maximization problems with a convex or quasiconvex objective function. As an additional corollary, we show that the number of vertices in an integer hull is bounded by $$O(n)^n$$ O ( n ) n for $$\Delta = O(1)$$ Δ = O ( 1 ) . As a part of our method, we introduce the notion of deep bases of a linear program. The problem to estimate their number by a non-trivial way seems to be quite challenging.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:snopef:v:5:y:2024:i:2:d:10.1007_s43069-024-00310-2
    DOI: 10.1007/s43069-024-00310-2
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