Proximity and Flatness Bounds for Linear Integer Optimization
本文针对线性整数优化中的两个核心问题——最优整数解与线性松弛解的距离（邻近性界）以及无整数点多面体的整数平行超平面数（平坦性界）——提出了改进的上界，通过一种新的证明技术建立了二者之间的联系。
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 [Formula: see text] satisfies [Formula: see text], where τ denotes [Formula: see text] counterclockwise rotation and [Formula: see text] denotes the polar of K, then the area of [Formula: see text] 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.