Computing the Difficulty of Critical Bootstrap Percolation Models is NP-hard

Abstract

Bootstrap percolation is a class of cellular automata with random initial state. Two-dimensional bootstrap percolation models have three 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 combinatorial quantity called ‘difficulty’, so the subject seemed closed up to finding sharper results. In this paper we prove that computing the difficulty of a critical model is NP-hard and exhibit an algorithm to determine it, in contrast with the upcoming result of Balister, Bollobás, Morris and Smith on undecidability in higher dimensions. The proof of NP-hardness is achieved by a reduction to the Set Cover problem.

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