  We prove that integer programming with three quantifier alternations is
$NP$-complete, even for a fixed number of variables. This complements earlier
results by Lenstra and Kannan, which together say that integer programming with
at most two quantifier alternations can be done in polynomial time for a fixed
number of variables. As a byproduct of the proof, we show that for two
polytopes $P,Q \subset \mathbb{R}^4$ , counting the projection of integer
points in $Q \backslash P$ is $\#P$-complete. This contrasts the 2003 result by
Barvinok and Woods, which allows counting in polynomial time the projection of
integer points in $P$ and $Q$ separately.
