Proximity and Flatness Bounds for Linear Integer Optimization

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 $K\subseteq {\mathbb{R}}^2$ satisfies $\tau K\subseteq K°$, where τ denotes $90°$ counterclockwise rotation and $K°$ denotes the polar of K, then the area of $K°$ is at least three.

Funding: J. Paat was supported by the Natural Sciences and Engineering Research Council of Canada [Grant RGPIN-2021-02475]. R. Weismantel was supported by the Einstein Stiftung Berlin.