Given convex polytope $\mathcal P\subseteq\Bbb R^n$ with $\mathcal P_\Bbb Z\leq2^n$ integer points and given locations of $O(\log \mathcal P_\Bbb Z)$ integer points in some positions can we obtain $\mathcal P_\Bbb Z$ in $O(n^c)$ time at a $c>0$?
Does the location of revealed points affect the complexity?
Intuitively, specifying a small number of integer points in a small region of some polytope has almost no connection with the rest of the polytope, so the answer should be negative. To give a concrete example, let $P$ be an $n$-element poset and $\mathcal{O}(P)$ its order polytope (see http://math.mit.edu/~rstan/pubs/pubfiles/66.pdf). If you are given the facets of $\mathcal{O}(P)$ then computing the number of integer points is equivalent to computing the number $j(P)$ of order ideals of $P$ (which is at most $2^n$), and is known to be $\#P$-complete. Suppose $P$ is the disjoint union of an $n/2$-element antichain (say) and an $n/2$-element poset $Q$. Then we know the $(n/2)!$ order ideals of the antichain, which is far more than $\log_2 j(P)\leq n$. Moreover, $j(Q)=(n/2)!j(Q)$, and computing $j(Q)$ is $\#P$-complete. I doubt whether specifying any $n$ (or some much larger number) of order ideals of $P$ will be of much help in counting the total number of them.
