A Quantitative Doignon-Bell-Scarf Theorem

The famous Doignon-Bell-Scarf Theorem is a Helly-type result about the existence of integer solutions on systems of linear inequalities. The purpose of this paper is to present the following quantitative generalization: Given an integer $k$, we prove that there exists a constant $c(n,k)$, depending only on the dimension $n$ and $k$, such that if a polyhedron ${x: Ax \leq b}$ contains exactly k integer solutions, then there exists a subset of the rows, of cardinality no more than $c(n,k)$, defining a polyhedron that contains exactly the same $k$ integer points. In this case $c(n,0) = 2^n$ is the original case of Doignon-Bell-Scarf for infeasible systems of inequalities. We work on both upper and lower bounds for the constant $c(n,k)$ and discuss some consequences, including a Clarkson-style algorithm to find the $l$-th best solution of an integer program with respect to the ordering induced by the objective function.