Vaccinate your trees!

Ehard, Stefan; Rautenbach, Dieter

doi:10.1016/j.tcs.2018.11.018

Cited by 3 publications

(2 citation statements)

References 15 publications

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“…Centeno and Rautenbach [6] establish bounds for the problems considered in [17]. In [14], Ehard and Rautenbach consider the following two variants of (2) for a given triple (G, τ, b),…”

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confidence: 99%

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Partial immunization of trees

Dourado

Ehard

Penso

et al. 2020

Discrete Optimization

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For a graph G and an integer-valued function τ on its vertex set, a dynamic monopoly is a set of vertices of G such that iteratively adding to it vertices u of G that have at least τ (u) neighbors in it eventually yields the vertex set of G. We study the problem of maximizing the minimum order of a dynamic monopoly by increasing the threshold values of individual vertices subject to vertex-dependent lower and upper bounds, and fixing the total increase. We solve this problem efficiently for trees, which extends a result of Khoshkhah and Zaker (On the largest dynamic monopolies of graphs with a given average threshold, Canadian Mathematical Bulletin 58 (2015) 306-316). IntroductionAs a simple model for an infection process within a network [12,13,16] one can consider a graph G in which each vertex u is assigned a non-negative integral threshold value τ (u) quantifying how many infected neighbors of u are required to spread the infection to u. In this setting, a dynamic monopoly of (G, τ ) is a set D of vertices such that an infection starting in D spreads to all of G, and the smallest order dyn(G, τ ) of such a dynamic monopoly measures the vulnerability of G for the given threshold values.Khoshkhah and Zaker [17] consider the maximum of dyn(G, τ ) over all choices for the function τ such that the average threshold is at most some positive realτ . They show that this maximum equals max k :

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“…Centeno and Rautenbach [6] establish bounds for the problems considered in [17]. In [14], Ehard and Rautenbach consider the following two variants of (2) for a given triple (G, τ, b),…”

mentioning

confidence: 99%

“…Our approach to show Theorem 1.1 is similar as in [14] and relies on recursive expressions for the following two quantities: For a vertex u of T and a non-negative integer b, let…”

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confidence: 99%

Partial immunization of trees

Dourado

Ehard

Penso

et al. 2020

Discrete Optimization

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Parameterized Complexity of Immunization in the Threshold Model

Cordasco¹,

Gargano²,

Rescigno³

2022

WALCOM: Algorithms and Computation

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Immunization in the Threshold Model: A Parameterized Complexity Study

2023

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We consider the problem of keeping under control the spread of harmful items in networks, such as the contagion proliferation of diseases or the diffusion of fake news. We assume the linear threshold model of diffusion where each node has a threshold that measures the node’s resistance to the contagion. We study the parameterized complexity of the problem: Given a network, a set of initially contaminated nodes, and two integers k and $$\ell $$ ℓ , is it possible to limit the diffusion to at most k other nodes of the network by immunizing at most $$\ell $$ ℓ nodes? We consider several parameters associated with the input, including the bounds k and $$\ell $$ ℓ , the maximum node degree $$\Delta $$ Δ , the number $$\zeta $$ ζ of initially contaminated nodes, the treewidth, and the neighborhood diversity of the network. We first give W[1] or W[2]-hardness results for each of the considered parameters. Then we give fixed-parameter algorithms for some parameter combinations.

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Vaccinate your trees!

Cited by 3 publications

References 15 publications

Partial immunization of trees

Partial immunization of trees

Parameterized Complexity of Immunization in the Threshold Model

Immunization in the Threshold Model: A Parameterized Complexity Study

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