We give an efficient perfect sampling algorithm for weighted, connected induced subgraphs (or graphlets) of rooted, bounded degree graphs. Our algorithm utilizes a vertex-percolation process with a carefully chosen rejection filter and works under a percolation subcriticality condition. We show that this condition is optimal in the sense that the task of (approximately) sampling weighted rooted graphlets becomes impossible in finite expected time for infinite graphs and intractable for finite graphs when the condition does not hold. We apply our sampling algorithm as a subroutine to give near linear-time perfect sampling algorithms for polymer models and weighted non-rooted graphlets in finite graphs, two widely studied yet very different problems. This new perfect sampling algorithm for polymer models gives improved sampling algorithms for spin systems at low temperatures on expander graphs and unbalanced bipartite graphs, among other applications.

翻译：我们提出了一种高效的完美采样算法，用于对根节点有界度图中的加权连通诱导子图（或图元）进行采样。该算法利用顶点渗透过程并精心选择拒绝滤波器，在渗透亚临界条件下运行。我们证明这一条件是最优的，即当该条件不成立时，对无限图而言（近似）采样加权根图元在有限期望时间内不可能实现，对有限图而言则难以处理。我们将该采样算法作为子程序，应用于有限图中的聚合物模型和加权无根图元的近乎线性时间完美采样，这两个问题被广泛研究却性质迥异。这种针对聚合物模型的新型完美采样算法，为扩展图和不平衡二分图上的低温自旋系统等应用提供了改进的采样方法。