On the number of integer points in translated and expanded polyhedra

TL;DR

This paper proves that minimizing integer points in translated and expanded rational convex polytopes in six dimensions is NP-hard, highlighting the computational difficulty and complex behavior of Ehrhart quasi-polynomials.

Contribution

It establishes NP-hardness results for problems related to integer points in polytope translations and expansions in six dimensions, revealing complexity in Ehrhart polynomial fluctuations.

Findings

Minimizing integer points in translated polytopes is NP-hard in 6D.

Finding maximal expansion with limited integer points is NP-hard.

Ehrhart quasi-polynomials can exhibit arbitrary fluctuations.

Abstract

We prove that the problem of minimizing the number of integer points inparallel translations of a rational convex polytope in $\mathbb{R}^6$ is NP-hard. We apply this result to show that given a rational convex polytope $P \subset \mathbb{R}^6$, finding the largest integer $t$ s.t. the expansion $tP$ contains fewer than $k$ integer points is also NP-hard. We conclude that the Ehrhart quasi-polynomials of rational polytopes can have arbitrary fluctuations.