  Given a distribution of pebbles to the vertices of a graph, a pebbling move
removes two pebbles from a single vertex and places a single pebble on an
adjacent vertex. The pebbling number $\pi(G)$ is the smallest number such that,
for any distribution of $\pi(G)$ pebbles to the vertices of $G$ and choice of
root vertex $r$ of $G$, there exists a sequence of pebbling moves that places a
pebble on $r$. Computing $\pi(G)$ is provably difficult, and recent methods for
bounding $\pi(G)$ have proved computationally intractable, even for moderately
sized graphs. Graham conjectured that $\pi(G ~\square~ H) \leq \pi(G) \pi(H)$,
where $G ~\square~ H$ is the Cartesian product of $G$ and $H$ (1989). While the
conjecture has been verified for specific families of graphs, in general it
remains open. This study combines the focus of developing a computationally
tractable, IP-based method for generating good bounds on $\pi(G ~\square~ H)$,
with the goal of shedding light on Graham's conjecture.We provide computational
results for a variety of Cartesian-product graphs, including some that are
known to satisfy Graham's conjecture and some that are not. Our approach leads
to a sizable improvement on the best known bound for $\pi(L ~\square~ L)$,
where $L$ is the Lemke graph, and $L ~\square~ L$ is among the smallest known
potential counterexamples to Graham's conjecture.
