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.

我们为有根、有界度图的加权、连通归纳子图（或小图）提供了一种高效的完美采样算法。我们的算法利用带有精心选择的拒绝滤波器的顶点渗滤过程，并在渗滤亚临界条件下工作。我们表明，这种条件是最优的，因为在无限图的有限预期时间内，（近似）采样加权有根图的任务变得不可能，并且当条件不成立时，对于有限图来说，采样任务变得棘手。我们将采样算法作为子程序，为聚合物模型和有限图中的加权无根图提供近线性时间完美采样算法，这是两个广泛研究但又截然不同的问题。这种用于聚合物模型的新的完美采样算法为扩展图和不平衡二部图等应用中的低温下自旋系统提供了改进的采样算法。