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Proximity and Flatness Bounds for Linear Integer Optimization

Author

Listed:
  • Marcel Celaya

    (School of Mathematics, Cardiff University, Wales CF24 4AG, United Kingdom)

  • Stefan Kuhlmann

    (Institut für Mathematik, Technische Universität Berlin, 10587 Berlin, Germany)

  • Joseph Paat

    (Sauder School of Business, University of British Columbia, Vancouver, British Columbia V6T 1ZC, Canada)

  • Robert Weismantel

    (Department of Mathematics, Institute for Operations Research, Eidgenössische Technische Hochschule Zürich, 8092 Zurich, Switzerland)

Abstract
This paper deals with linear integer optimization. We develop a technique that can be applied to provide improved upper bounds for two important questions in linear integer optimization. Given an optimal vertex solution for the linear relaxation, how far away is the nearest optimal integer solution (if one exists; proximity bounds)? If a polyhedron contains no integer point, what is the smallest number of integer parallel hyperplanes defined by an integral, nonzero, normal vector that intersect the polyhedron (flatness bounds)? This paper presents a link between these two questions by refining a proof technique that has been recently introduced by the authors. A key technical lemma underlying our technique concerns the areas of certain convex polygons in the plane; if a polygon K ⊆ R 2 satisfies τ K ⊆ K ° , where τ denotes 90 ° counterclockwise rotation and K ° denotes the polar of K , then the area of K ° is at least three.

Suggested Citation

  • Marcel Celaya & Stefan Kuhlmann & Joseph Paat & Robert Weismantel, 2024. "Proximity and Flatness Bounds for Linear Integer Optimization," Mathematics of Operations Research, INFORMS, vol. 49(4), pages 2446-2467, November.
    • Handle: RePEc:inm:ormoor:v:49:y:2024:i:4:p:2446-2467
    DOI: 10.1287/moor.2022.0335
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