Sublinear Bounds for a Quantitative Doignon-Bell-Scarf Theorem

TL;DR

This paper improves bounds on the number of inequalities needed to describe polyhedra with exactly k integer points, showing a sublinear relationship in k, and provides structural insights and tight bounds for dimension two.

Contribution

It establishes a sublinear asymptotic bound for the function c(n,k), improving previous exponential bounds, and offers lower bounds and structural results, especially tight bounds in two dimensions.

Findings

Bound c(n,k) = O(k^{1- heta}) for some >0, improving previous exponential bounds.

Provides lower bounds on c(n,k), showing the bounds are nearly tight.

Achieves asymptotically tight bounds for the case n=2.

Abstract

The recent paper "A quantitative Doignon-Bell-Scarf Theorem" by Aliev et al. generalizes the famous Doignon-Bell-Scarf Theorem on the existence of integer solutions to systems of linear inequalities. Their generalization examines the number of facets of a polyhedron that contains exactly $k$ integer points in $\mathbb{R}^n$. They show that there exists a number $c(n,k)$ such that any polyhedron in $\mathbb{R}^n$ that contains exactly $k$ integer points has a relaxation to at most $c(n,k)$ of its inequalities that will define a new polyhedron with the same integer points. They prove that $c(n,k) = O(k2^n)$. In this paper, we improve the bound asymptotically to be sublinear in $k$. We also provide lower bounds on $c(n,k)$, along with other structural results. For dimension $n=2$, our bounds are asymptotically tight to within a constant.