The maximum time of 2-neighbour bootstrap percolation: algorithmic aspects

Benevides, Fabrício Siqueira; Campos, Víctor; Dourado, Mitre C.; Sampaio, Rudini Menezes; Silva, Ana

doi:10.1007/978-88-7642-475-5_22

“…Analogously we deal with the fact that construction (7) does not apply for k = 4. Thus the lower bound on M (k, ℓ) is proved.…”

Section: Letmentioning

confidence: 99%

“…Assume that M (R ′ 1 ) ≥ M (R ′ 2 ). If R ′ 1 and R ′ 2 satisfy Condition F inside R ′ , with R ′ 2 being a single site, then we can bound the (total) time that A takes to percolate in a much better way than using inequality (7). In fact, considering the possible cases it can, again, be bounded from above by (5).…”

Section: Proof Of Claimmentioning

confidence: 99%

Maximum Percolation Time in Two-Dimensional Bootstrap Percolation

Benevides¹,

Przykucki²

2015

SIAM J. Discrete Math.

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We consider a classic model known as bootstrap percolation on the n × n square grid. To each vertex of the grid we assign an initial state, infected or healthy, and then in consecutive rounds we infect every healthy vertex that has at least two already infected neighbors. We say that percolation occurs if the whole grid is eventually infected. In this paper, contributing to a recent series of extremal results in this field, we prove that the maximum time a bootstrap percolation process can take to eventually infect the entire vertex set of the grid is 13n 2 /18 + O(n).

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“…In 2013, Benevides et.al. [10], among other results, proved that the Percolation Time Problem is polynomial time solvable for k ≤ 2, but is NP-complete for k ≥ 4 and, when restricted to bipartite graphs, it is NP-complete for k ≥ 7. Moreover, it was proved that the Percolation Time Problem is NP-complete for planar graphs.…”

Section: Percolation Timementioning

confidence: 99%

“…In this paper, we solve the third question of [10]. We prove that the Percolation Time Problem is NPcomplete for planar bipartite graphs.…”

Section: Percolation Timementioning

confidence: 99%

The Maximum Time of 2-neighbour Bootstrap Percolation in Grid Graphs and Parametrized Results

Marcilon

¹

,

Sampaio

²

2016

Graph-Theoretic Concepts in Computer Science

0

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In 2-neighborhood bootstrap percolation on a graph G, an infection spreads according to the following deterministic rule: infected vertices of G remain infected forever and in consecutive rounds healthy vertices with at least two already infected neighbors become infected. Percolation occurs if eventually every vertex is infected. The maximum time t(G) is the maximum number of rounds needed to eventually infect the entire vertex set. In 2013, it was proved by Benevides et al [10] that t(G) is NP-hard for planar graphs and that deciding whether t(G) ≥ k is polynomial time solvable for k ≤ 2, but is NP-complete for k ≥ 4. They left two open problems about the complexity for k = 3 and for planar bipartite graphs. In 2014, we solved the first problem [28]. In this paper, we solve the second one by proving that t(G) is NP-complete even in grid graphs with maximum degree 3. We also prove that t(G) is polynomial time solvable for solid grid graphs with maximum degree 3. Moreover, we prove that the percolation time problem is W[1]-hard on the treewidth of the graph, but it is fixed parameter tractable with parameters treewidth+k and maxdegree+k.

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“…In [7], it was proved that deciding if t(G) ≥ 7 is NP-Complete in bipartite graphs. The following theorem improves this result.…”

Section: T(g) ≥ 5 Is Np-complete In Bipartite Graphsmentioning

confidence: 99%

The maximum time of 2-neighbor bootstrap percolation: Complexity results

Marcilon¹,

Sampaio²

2018

Theoretical Computer Science

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In 2-neighborhood bootstrap percolation on a graph G, an infection spreads according to the following deterministic rule: infected vertices of G remain infected forever and in consecutive rounds healthy vertices with at least 2 already infected neighbors become infected. Percolation occurs if eventually every vertex is infected. The maximum time t(G) is the maximum number of rounds needed to eventually infect the entire vertex set. In 2013, it was proved [7] that deciding whether t(G) ≥ k is polynomial time solvable for k = 2, but is NP-Complete for k = 4 and, if the problem is restricted to bipartite graphs, it is NP-Complete for k = 7. In this paper, we solve the open questions. We obtain an O(mn 5 )time algorithm to decide whether t(G) ≥ 3. For bipartite graphs, we obtain an O(mn 3 )-time algorithm to decide whether t(G) ≥ 3, an O(m 2 n 9 )-time algorithm to decide whether t(G) ≥ 4 and we prove that t(G) ≥ 5 is NP-Complete.Keywords: 2-neighbor bootstrap percolation, P 3 -Convexity, maximum time, infection on graphs t r (G, S, v) = ∞ if there is no such t). We say that a set S (0) infects G (or that S (0) is a percolating set of G) if eventually every vertex of G becomes infected, that is, there exists t such that S (t) = V (G). If S is a percolating set of G, then we define t r (G, S) as the minimum t such that S (t) = V (G). Also, define the percolation time of G as t r (G) = max{t r (G, S) : S infects G}. In this paper, we shall focus on the case where r = 2 and in such case we omit the subscript of the functions t r (G, S, v), t r (G, S) and t r (G).Bootstrap percolation was introduced by Chalupa, Leath and Reich [15] as a model for certain interacting particle systems in physics. Since then it has found applications in clustering phenomena, sandpiles [22], and many other areas of statistical physics, as well as in neural networks [1] and computer science [18].There are two broad classes of questions one can ask about bootstrap percolation. The first, and the most extensively studied, is what happens when the initial configuration S (0) is chosen randomly under some probability distribution? For example, vertices are included in S (0) independently with some fixed probability p. One would like to know how likely percolation is to occur, and if it does occur, how long it takes.The answer to the first of these questions is now well understood for various graphs. An interesting case is the one of the lattice graph [n] d , in which d is fixed and n tends to infinity, since the probability of percolation under the r-neighbor model displays a sharp threshold between no percolation with high probability and percolation with high probability. The existence of thresholds in the strong sense just described first appeared in papers by Holroyd, Balogh, Bollobás, Duminil-Copin and Morris [24,5,3]. Sharp thresholds have also been proved for the hypercube (Balogh and Bollobás [2], and Balogh, Bollobás and Morris [6]). There are also very recent results due to Bollobás, Holmgren, Smith and Uzzell [11], about the time perco...

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The maximum time of 2-neighbour bootstrap percolation: algorithmic aspects

Cited by 11 publications

References 17 publications

Maximum Percolation Time in Two-Dimensional Bootstrap Percolation

Maximum Percolation Time in Two-Dimensional Bootstrap Percolation

The Maximum Time of 2-neighbour Bootstrap Percolation in Grid Graphs and Parametrized Results

The maximum time of 2-neighbor bootstrap percolation: Complexity results

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