  We present an efficient reduction from the Bounded integer programming (BIP)
to the Subspace avoiding problem (SAP) in lattice theory. The reduction has
some special properties with some interesting consequences. The first is the
new upper time bound for BIP, $poly(\varphi)\cdot n^{n+o(n)}$ (where $n$ and
$\varphi$ are the dimension and the input size of the problem, respectively).
This is the best bound up to now for BIP. The second consequence is the proof
that #SAP, for some norms, is #P-hard under semi-reductions. It follows that
the counting version of the Generalized closest vector problem is also #P-hard
under semi-reductions. Furthermore, we also show that under some reasonable
assumptions, BIP is solvable in probabilistic time $2^{O(n)}$.
