SIAM Journal on Discrete Mathematics, Volume 34, Issue 2, Page 1444-1459, January 2020.
Bootstrap percolation is a class of cellular automata with random initial state. Two-dimensional bootstrap percolation models have three rough universality classes, the most studied being the “critical” one. For this class the scaling of the quantity of greatest interest (the critical probability) was determined by Bollobás, Duminil-Copin, Morris, and Smith in terms of a simply defined combinatorial quantity called “difficulty,” so the subject seemed closed up to finding sharper results. However, the computation of the difficulty was never considered. In this paper we provide the first algorithm to determine this quantity, which is, surprisingly, not as easy as the definition leads to thinking. The proof also provides some explicit upper bounds, which are of use for bootstrap percolation. On the other hand, we also prove the negative result that computing the difficulty of a critical model is NP-hard. This two-dimensional picture contrasts with an upcoming result of Balister, Bollobás, Morris, and Smith on uncomputability in higher dimensions. The proof of NP-hardness is achieved by a technical reduction to the Set Cover problem.

中文翻译：

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二维Bootstrap渗透困难的复杂性：算法和NP-Hardness

SIAM离散数学杂志，第34卷，第2期，第1444-1459页，2020年1月。
自举渗滤是一类具有随机初始状态的细胞自动机。二维自举渗透模型具有三个粗糙的通用性类别，其中研究最多的是“关键”类别。对于此类，最大的兴趣量（临界概率）的缩放比例是由Bollobás，Duminil-Copin，Morris和Smith决定的，其定义为一个简单的组合量，称为“难度”，因此该主题似乎不愿寻找更清晰的结果。但是，从未考虑过难度的计算。在本文中，我们提供了确定该数量的第一个算法，令人惊讶的是，它不像定义所导致的那样容易。该证明还提供了一些明确的上限，可用于引导渗透。另一方面，我们还证明了计算关键模型的难度是NP-hard的负面结果。这张二维图片与Balister，Bollobás，Morris和Smith即将发表的关于较高维度的不可计算性的结果形成对比。NP硬度的证明是通过技术上减少了Set Cover问题来实现的。