  Graph pebbling is a combinatorial game played on an undirected graph with an
initial configuration of pebbles. A pebbling move consists of removing two
pebbles from one vertex and placing one pebble on an adjacent vertex. The
pebbling number of a graph is the smallest number of pebbles necessary such
that, given any initial configuration of pebbles, at least one pebble can be
moved to a specified root vertex. Recent lines of inquiry apply computational
techniques to pebbling bound generation and improvement.
  Along these lines, we present a computational framework that produces a set
of tree strategy weight functions that are capable of proving pebbling number
upper bounds on a connected graph. Our mixed-integer linear programming
approach automates the generation of large sets of such functions and provides
verifiable certificates of pebbling number upper bounds. The framework is
capable of producing verifiable pebbling bounds on any connected graph,
regardless of its structure or pebbling properties. We apply the model to the
4th weak Bruhat to prove $\pi(B_4) \leq 66$ and to the Lemke square graph to
produce a set of certificates that verify $\pi(L x L) \leq 96$.
