Abstract. We show that for every [Formula: see text], the maximal running time of the [Formula: see text]-bootstrap percolation in the complete [Formula: see text]-uniform hypergraph on [Formula: see text] vertices [Formula: see text] is [Formula: see text]. This answers a recent question of Noel and Ranganathan in the affirmative and disproves a conjecture of theirs. Moreover, we show that the prefactor is of the form [Formula: see text] as [Formula: see text].

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.