Complexity of Two-dimensional Bootstrap Percolation Difficulty: Algorithm and NP-Hardness

Abstract

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 [5] 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 [3] on uncomputability in higher dimensions. The proof of NP-hardness is achieved by a technical reduction to the Set Cover problem.

Universality for critical KCM: infinite number of stable directions

“…A quantitative version of this result was proved by Mezei and the first author [21]. An easy corollary of Observation 5.13 is the fact that crumbs can only grow very locally (see Figure 5a).…”

“…Furthermore (see [7,Lemma 2.7], [8, Lemma 5.2]), 1 ď αpuq ă 8 if and only if u is an isolated stable direction, so that U is critical if and only if 1 ď αpUq ă 8. As a final remark we recall that, contrary to determining whether an update family is critical, finding αpUq is a NP-hard question [21].…”