Let $P$ be a polytope in $\mathbb{R}^n$. Let $A_ix = b_i$ be the defining equations of its codimension $1$ faces. Is there an algorithm or some kind of criterion to decide if the lattice points inside $P$ and on its boundary have the property that any two are connected by a lattice path through $P$ (and its boundary)? It seems to me that this is a hard problem in general even for small $n$.

In the above, the lattice in question is not mentioned on purpose because I'm really looking for any kind of result. But I will certainly be happy for results concerning $\mathbb{Z}^n$.