Minimizing the number of lattice points in a translated polygon

Abstract

The parametric lattice-point counting problem is as follows: Given an integer matrix $A \in Z^{m \times n}$, compute an explicit formula parameterized by $b \in R^m$ that determines the number of integer points in the polyhedron $\{x \in R^n : Ax \leq b\}$. In the last decade, this counting problem has received considerable attention in the literature. Several variants of Barvinok's algorithm have been shown to solve this problem in polynomial time if the number $n$ of columns of $A$ is fixed. Central to our investigation is the following question: Can one also efficiently determine a parameter $b$ such that the number of integer points in $\{x \in R^n : Ax \leq b\}$ is minimized? Here, the parameter $b$ can be chosen from a given polyhedron $Q \subseteq R^m$. Our main result is a proof that finding such a minimizing parameter is $NP$-hard, even in dimension 2 and even if the parametrization reflects a translation of a 2-dimensional convex polygon. This result is established via a relationship of this problem to arithmetic progressions and simultaneous Diophantine approximation. On the positive side we show that in dimension 2 there exists a polynomial time algorithm for each fixed $k$ that either determines a minimizing translation or asserts that any translation contains at most $1 + 1/k$ times the minimal number of lattice points.