Integer decomposition property of dilated polytopes

Let $\mathcal{P}\subset {\mathbb{R}}^N$ be an integral convex polytope of dimension $d$ and write $k\mathcal{P}$, where $k=1,2,\dots$, for dilations of $\mathcal{P}$. We say that $\mathcal{P}$ possesses the integer decomposition property if, for any integer $k=1,2,\dots$ and for any $\alpha \in k\mathcal{P}\cap {\mathbb{Z}}^N$, there exist $\alpha_{1}, \ldots, \alpha_k$belongingto$$\mathcal{P}$ \cap $\mathbb{Z}$^N$suchthat$\alpha = \alpha_1 + \cdots + \alpha_k$. A fundamental question is to determine the integers $k>0$ for which the dilated polytope $k\mathcal{P}$ possesses the integer decomposition property. In the present paper, combinatorial invariants related to the integer decomposition property of dilated polytopes will be proposed and studied.