Proximity and flatness bounds for linear integer optimization

TL;DR

This paper introduces improved bounds for proximity and flatness in linear integer optimization, linking geometric properties of convex polygons to optimization bounds, thus advancing theoretical understanding.

Contribution

The authors refine a proof technique to establish tighter bounds for proximity and flatness, connecting geometric polygon properties to optimization problems.

Findings

Established a lower bound of 3 on the area of certain convex polygons' polars.

Linked proximity bounds to flatness bounds through geometric analysis.

Provided a novel proof technique refining previous methods.

Abstract

We develop a technique that can be applied to provide improved upper bounds for two important questions in linear integer optimization. - Proximity bounds: Given an optimal vertex solution for the linear relaxation, how far away is the nearest optimal integer solution (if one exists)? - Flatness bounds: If a polyhedron contains no integer point, what is the smallest number of integer parallel hyperplanes defined by an integral, non-zero, normal vector that intersect the polyhedron? 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 $\tau K \subseteq K^{\circ}$, where $\tau$ denotes $90^{\circ}$ counterclockwise rotation and $K^{\circ}$ denotes the polar of $K$, then the area of $K^{\circ}$ is at least 3.