  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 Bollob\'as, 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, Bollob\'as, Morris and Smith on uncomputability in higher dimensions.
The proof of NP-hardness is achieved by a technical reduction to the Set Cover
problem.
