  We consider so called $2$-stage stochastic integer programs (IPs) and their
generalized form of multi-stage stochastic IPs. A $2$-stage stochastic IP is an
integer program of the form $\max \{ c^T x \mid Ax = b, l \leq x \leq u, x \in
\mathbb{Z}^{nt + s} \}$ where the constraint matrix $A \in \mathbb{Z}^{r \times
s}$ consists roughly of $n$ repetition of a block matrix $A$ on the vertical
line and $n$ repetitions of a matrix $B \in \mathbb{Z}^{r \times t}$ on the
diagonal. In this paper we improve upon an algorithmic result by Hemmecke and
Schultz form 2003 to solve $2$-stage stochastic IPs. The algorithm is based on
the Graver augmentation framework where our main contribution is to give an
explicit doubly exponential bound on the size of the augmenting steps. The
previous bound for the size of the augmenting steps relied on non-constructive
finiteness arguments from commutative algebra and therefore only an implicit
bound was known that depends on parameters $r,s,t$ and $\Delta$, where $\Delta$
is the largest entry of the constraint matrix. Our new improved bound however
is obtained by a novel theorem which argues about the intersection of paths in
a vector space. As a result of our new bound we obtain an algorithm to solve
$2$-stage stochastic IPs in time $poly(n,t) \cdot f(r,s,\Delta)$, where $f$ is
a doubly exponential function. To complement our result, we also prove a doubly
exponential lower bound for the size of the augmenting steps.
