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 $\pi(L x L) \leq 96$.

翻译：图铺砌是一种在无向图上进行的组合游戏，其初始配置有若干铺砌块。一次铺砌操作包括从一个顶点移除两个铺砌块，并在相邻顶点放置一个铺砌块。图的铺砌数是指在任意初始铺砌配置下，至少有一个铺砌块能被移动至指定根顶点所需的最少铺砌块数量。近年来的研究方向将计算技术应用于铺砌数界的生成与优化。沿着这一思路，我们提出了一种计算框架，该框架能生成一组树策略权重函数，用于证明连通图的铺砌数上界。我们的混合整数线性规划方法可自动化生成大规模此类函数，并提供可验证的铺砌数上界证明。该框架能够为任意连通图生成可验证的铺砌数上界，无论其结构或铺砌性质如何。我们将该模型应用于第四弱布鲁哈特图，证明 $\pi(B_4) \leq 66$；并应用于莱姆克平方图，生成一组证明 $\pi(L \times L) \leq 96$ 的证书。