Bootstrap percolation is a class of cellular automata whose first representative was introduced in 1979 by Chalupa, Leath and Reich  in statistical physics. Further applications to several other areas have been considered, namely dynamics of the Ising model, kinetically constrained models for the glass transition, abelian sandpiles and others (see a recent review of Morris  for more information). Given a set or of initially infected sites, more vertices become infected at each discrete time step following a deterministic monotone local rule invariant in time and space, while infections never heal. More precisely, let us introduce the broadest framework introduced by Bollobás, Smith and Uzzell .111Earlier partly non-rigorous considerations of a more restricted class of models can be found in the works of Gravner and Griffeath [10, 11] from the 1990s.
A bootstrap percolation model is given by a finite set , called the update family, of finite subsets of , called rules. For an initial set of infected sites we define
and is the closure of with respect to this operation. We will only discuss the most studied case, where is taken at random according to the product Bernoulli measure , so that each site is initially infected with probability
. Equipped with this measure, the model exhibits a phase transition at
The model is defined identically on tori by setting
Although for some concrete models higher dimensions have been understood and some general universality conjectures have been put forward in [2, Conjecture 16] and [4, Conjecture 9.2], we will restrict our attention to the 2-dimensional case. The results of Bollobás, Smith and Uzzell  and Balister, Bollobás, Przykucki and Smith  combined establish that all bootstrap percolation models can be partitioned (by a simple procedure) into “universality classes” with the similar scaling of . In order to define these we need some notation. For a direction we denote by
the open half-plane with normal and by
the line passing through perpendicular to . A direction is unstable if there exists such that and stable otherwise. It is not difficult to show (see ) that the unstable directions form a finite union of open intervals in ; the partition into universality classes conjectured in  and proved in [6, 2] is in terms of these directions.
is supercritical if there exists an open semi-circle of unstable directions, in which case .
is critical if it is not supercritical and there exists a semi-circle with a finite number of stable directions, in which case .
is subcritical otherwise (if each semi-circle contains infinitely many stable directions), in which case .
The behavior of supercritical models is dominated by the study of finite infected sets with infinite closure, while subcritical ones are more closely related to percolation. The most studied models are critical ones, to which the archetypal example of bootstrap percolation belongs — the -neighbor model, in which a site becomes infected if at least two of its nearest neighbors are already infected. This is the first model for which the universality result above (and more) was established — by Aizenman and Lebowitz . They realized that the dynamics is dominated by a bottleneck — creating an infected “droplet” of a certain “critical” size, which can then easily grow out to infinity, and proved that for this model . In a substantial breakthrough Holroyd  determined the asymptotic location of the sharp threshold and since then much sharper results have been proved [12, 14]:
Such sharp or sharper bounds have been obtained for a handful of other specific models [9, 5, 8], but still remain open in general. However, the level of precision of the Aizenman-Lebowitz result was established in full generality for critical models by Bollobás, Duminil-Copin, Morris and Smith . They introduce the following key notion of “difficulty”.
Definition 1.1 (Definition 1.2 of 222The definition we give is formally different from the one in , but the two are easily seen to be equivalent.).
Let be a critical model and be an isolated stable direction. We then define the difficulty of , , to be the minimum cardinality of a set such that is infinite. For unstable directions we set and for non-isolated stable ones we set . The difficulty of is
where is the set of open semi-circles of .
The result of  then states333They actually give matching bounds up to a constant factor, which requires dividing critical models into two subclasses with different logarithmic factors.
Motivated by the notion of critical densities, corresponding to difficulties, but for subcritical models, introduced by the first author in , which are rather complicated in nature, we examine how hard it is to actually determine given the update family. Somewhat surprisingly, given the simple Definition 1.1, it turns out that difficulties are difficult to determine. Another important motivation comes from a related phenomenon in higher dimensions noticed by Balister, Bollobás, Morris and Smith  and announced prior to our work. There, an even more striking result emerges owing to the richness of supercritical -dimensional models: the exponent determining the scaling of is uncomputable. In two dimensions we prove the following less deterring result.
The problem of computing the difficulty of a critical bootstrap percolation family is NP-hard.
The same result holds for the bilateral difficulty introduced by Martinelli, Morris and Toninelli  relevant for the kinetically constrained model associated to a critical bootstrap percolation model. Indeed, it suffices to consider the symmetrised family with from the proof of the theorem and proceed as in Section 2. Then one obtains .
This result is proved by a fairly technical reduction to the Set Cover decision problem in Section 2. However, in order for this result to be meaningful, we need to make sure that there is some algorithm to compute the difficulties. This is all the more necessary in view of the result of , stating that in higher dimensions this is not the case. Thus, it is important to have some bound on the complexity of the problem, as provided by the next theorem. The algorithm in question is fully explicit and given in Section 3.
There exists an algorithm which, given a critical bootstrap percolation family , computes its difficulty .444This result was proved independently by Balister, Bollobás, Morris and Smith .
In fact, it is not hard to check that our algorithm runs in time
where is defined in (4). This bound is clearly as sharp as a bound in terms of only can be, since the input can be that large.
2 NP-hardness: proof of Theorem 1.2
In this section we prove Theorem 1.2 by providing a reduction from Set Cover to 2D Critical-Bootstrap Difficulty. For the Set Cover problem we consider a universe and a collection of subsets of the universe and assume that and . The Set Cover problem asks for determining the minimum cardinality of a subset of which covers the universe. It is one of the first NP-complete problems described by Karp .
We fix an instance
Our goal is to define a critical bootstrap percolation family whose difficulty is (up to a simple transformation) the solution to Set Cover. Let the set of rules associated to be
and the rules defined as follows share a large portion of their structure (see Figure 1).
It is easy to check that the only isolated stable direction is , while are the only other stable ones, so that is critical and . We thus focus on this direction. Let be an optimal solution to the Set Cover problem given by i.e. a set of minimal size such that
Indeed, using the rules for and all , one infects all sites in
since is a cover, and those are enough to infect using and .
For any we denote
To prove that (2) is actually an equality, we suppose that there exists a set for which and . Fix a minimal such set . If there exists such that one of and is a subset of , then we have a contradiction with the assumption that . However, some of the rules must be applicable to and therefore there exists such that .
For any we have .
Although the verification is immediate, calling this fact an observation is deceptive, since is designed to possess this property. It follows that is unique, otherwise (since any minimal cover is smaller than the universe), a contradiction.
Every point has the same -coordinate as .
Suppose that there exists contradicting the statement of the lemma and consider such a with minimal infection time. Then contains at least sites on the row of . Therefore, , a contradiction. ∎
By monotonicity and Lemma 2.2, we can assume that (as long as we can replace by and the problem is invariant under translation by ).
The family is a cover of .
By the minimality of and Lemma 2.2, the -coordinate of any site in is 0, 1, or 2. Suppose that contains for some and take such that is the first such translate to become infected. By Lemma 2.2 is of the form .
If , then by Lemma 2.2 the set contains at least elements (with -coordinate ), therefore , a contradiction.
If , then either some sites in lying on the -axis have been infected via rule or , in which case it is already guaranteed that infinitely many sites become infected during the bootstrap process, or by of Observation 2.1, a contradiction. Thus, removing from every site in with -coordinate 1 does not prevent the infection of infinitely many sites, which contradicts the minimality of .
Hence, such a vectorcannot exist, so that until a rule or is used the only possible infections are of the form “ becomes infected via rule ”. Therefore, all sites are either redundant (which contradicts the minimality of ) or satisfy with .
Finally, set and assume that
Then, in order to have , it is necessary (and sufficient) to have a sequence of consecutive sites in
However, such a sequence is either disjoint from the infections of the form , in which case , or disjoint from . In the latter case the sequence contains at most
infected sites. This contradiction completes the proof. ∎
3 Decidability: proof of Theorem 1.4
In this section we provide an algorithm to compute the difficulty of a critical model. Let us stress that it is not optimized and is only meant to prove Theorem 1.4.
Proof of Theorem 1.4.
Fix a critical family . To start, let us note that the stable directions are trivially determined (in polynomial time) and there are at most isolated stable directions, so it suffices to show that one can compute the difficulty of a given isolated stable direction, since deducing the global difficulty of the model from directional ones is also easy by (1). Let us fix an isolated stable direction to consider and set
which we shall assume to be sufficiently large throughout the proof. Indeed, there are only a finite number of update families with smaller than a fixed value. Therefore, if one has an algorithm to compute for all families with larger , one can preset the output of the algorithm for those families to obtain an algorithm determining for any value of without altering the asymptotic complexity.
Recall the notation (3), which we shall use without specifying , as it will be clear from the context. In order to determine we will use the following lemmas to bound the size of the set in Definition 1.1. The first of these is a one-dimensional result which we shall reduce the problem to.
Let be a bootstrap family, let be an isolated stable direction and let . Then the set is either infinite or its maximal distance from is at most .
Observe that by stability of we have , so the dynamics (with fully infected) can be replaced by a one-dimensional bootstrap family acting on , that we identify with , so that becomes simply . Since is an isolated stable direction defined by , distances in are at most times larger than those in its identification with .
Denote with . Let us denote by the property that , for all , and . Also let be minimal not satisfying . We aim to prove that , so we assume the contrary.
We first note that , since, if a single site creates an additional infection, it necessarily creates an infinite arithmetic progression of infections. Assume that there exists and such that and . Then by minimality of both and satisfy . Therefore,
so that , which contradicts the existence of .
Assume next that (the corresponding case for is treated identically). Then, by the pigeon-hole principle, there exist with such that
(since no infection can cross a region of size not intersecting to reach ). Therefore, infects a translate of itself, since the dynamics to the right of is not affected by infections to the left of , once we fix the state of . This is a contradiction with , which concludes the proof. ∎
The next Lemma is an easy application of the covering algorithm of .
Let be a critical update family and be an isolated stable direction. Let be a set of size at most . Then for every we have .
We first claim that there exists a set of three or four stable directions containing the origin in their convex envelope such that for each there exists such that . Indeed, if is unstable, it suffices to take the stable directions closest to in both semi-circles ending at and . These directions satisfy the condition above as they are (semi-)isolated stable for and contain the origin in their convex envelope, since is not supercritical. If, on the contrary, is stable, we can pick any (semi-)isolated stable direction in both semi-circles ending at and or, if one of those circles is entirely stable, we take its midpoint. Adding and to those two stable directions, we obtain the desired .
Then observe that the angle between each two of these directions is (as the determinant of the integer points with those directions is a non-zero integer), so that there is a -droplet of diameter containing . We can then directly apply the covering algorithm of  to conclude the proof. Indeed, the covering algorithm applied to sites instead of breakthrough blocks starts with at most droplets of diameter and at each step the sum of the diameters of the remaining droplets increases by at most by Lemma 4.6 of , while the number of droplets decreases by . Hence, the output of the algorithm has total diameter . By Lemma 4.5 of  the output of the algorithm contains , so the proof is complete, as each of the output droplets contains at least one site of . ∎
Let us first describe an algorithm to determine and postpone its analysis. For each integer from to we successively perform the following operations to determine if there exists a set of size as in Definition 1.1. We stop as soon as such a set is found and return the corresponding (minimal) value of . For each fixed we start by choosing a set . The first site is and each new one is picked within distance from some of the previous ones and such that for some among the previous ones. There are at most
such choices. For each of them we successively inspect different translations , such that and (where is such that and and are co-prime), in the (total) order given by starting from . Finally, fix .
For each we run the bootstrap dynamics for until it either stops infecting new sites or infects a site with and . This can be done by checking at each step each site at distance from the origin for each rule and repeating this for time steps. If the dynamics becomes stationary, we continue, while otherwise we return for the value of .
We now turn to proving that the algorithm does return an output and it is precisely . The first assertion is easy. Indeed, as is an isolated stable direction, there exists a rule , so that adding consecutive sites on to is enough to infect a half-line of only taking into account. Thus, we know that and the algorithm will eventually check such a configuration when and infections will propagate to distance (and in fact to infinity). Let us then prove that the output is .
Assume that a set considered by the algorithm is of size such that is finite for all previous choices of including the current one. We prove by induction on that the maximal distance between a site from and is at most . Indeed, if , then and there is nothing to prove, since no additional infections take place. Assume the property to hold for all with in the order given by and denote and with the translate obtained for . Observe that by monotonicity for each we have that
for which the induction hypothesis applies. Thus, we only need to consider . However, by Lemma 3.1, sites there cannot reach distance more than from
which is at distance at most from itself, by the previous reasoning. This completes the induction (using that is independent of and is at least ) and the proof that the algorithm cannot return a value smaller than .
Finally, consider a set as in Definition 1.1 of size (and therefore minimal). Note that by minimality and Lemma 3.2, the projection of on cannot have a gap of length larger than . We also claim that its projection on cannot have a gap of length larger than , which suffices as any set satisfying these conditions is examined by the algorithm and by construction (and the previous reasoning) it will return the first one with infinite. Indeed, such a gap cannot exist by the reasoning from the previous paragraph applied to each of the two parts of separated by the gap. Thus, the output is indeed and the proof is complete. ∎
4 Open problems
Let us conclude with a few open questions naturally suggested by the present work. Of course, many more complexity issues arise systematically for hard problems, but let us mention the foremost ones.
Can one find a good approximation of in polynomial time?
Are there interesting subfamilies of critical models for which the difficulty is computable in polynomial time?
In view of Remark 1.5, can one find an algorithm whose complexity depends only on the size of the input , but not on the size of its entries ? Moreover, is bounded by a function of the input size and how large is such a function?
The authors would like to thank the organizers of ICGT 2018, Lyon, during which this project started. We also thank Rob Morris for helpful comments regarding .
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