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：我们提出了一种针对有根、有界度图上的加权连通诱导子图（或称图元）的高效完美采样算法。该算法利用精心选择的拒绝滤波器执行顶点渗流过程，并在渗流次临界条件下运行。我们证明，当该条件不成立时，对于无限图，（近似）采样加权有根图元的任务在有限期望时间内不可行，而对于有限图则变得棘手，因此该条件是最优的。我们将此采样算法作为子例程，为有限图上的聚合物模型和加权无根图元（两个广泛研究但截然不同的问题）提供了近线性时间的完美采样算法。这种针对聚合物模型的新型完美采样算法改进了低温条件下展开图与不平衡二分图上自旋系统的采样算法，并具有其他应用。