Graph Fragmentation Problem

Piccini, Juan; Robledo, Franco; Romero, Pablo

doi:10.5220/0005697701370144

Cited by 4 publications

(3 citation statements)

References 14 publications

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“…Indeed, in Piccini et al. () it is proved that a large set of node‐immunization problems are at least as hard as the GFP. Therefore, this proves that immunization is a hard task, and the intuition from epidemiologist is correct.…”

Section: Discussionmentioning

confidence: 99%

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Graph fragmentation problem: analysis and synthesis

Aprile

Castro

Ferreira

et al. 2018

Int Trans Operational Res

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Vulnerability metrics play a key role in the understanding of cascading failures and target/random attacks to a network. The graph fragmentation problem (GFP) is the result of a worst-case analysis of a random attack. We can choose a fixed number of individuals for protection, and a nonprotected target node immediately destroys all reachable nodes. The goal is to minimize the expected number of destroyed nodes in the network. In this paper, we address the GFP by several approaches: metaheuristics, approximation algorithms, polytime methods for specific instances, and exact methods for small instances. The computational complexity of the GFP is included in our analysis, where we formally prove that the corresponding decision version of the problem is N P-complete. Furthermore, a strong inapproximability result holds: there is no polynomial approximation algorithm with factor lower than 5/3, unless P = N P. This promotes the study of specific instances of the problem for tractability and/or exact methods in exponential time. As a synthesis, we propose new vulnerability/connectivity metrics and an interplay with game theory using a closely related combinatorial problem called component order connectivity.42 Aprile et al. / Intl. Trans. in Op. Res. 26 (2019) 41-53 system. More recently, the focus moved toward disaster management, centrality, and vulnerability metrics under random/targeted attacks (Thai and Pardalos, 2011;Mauthe et al., 2016;Gouveia and Leitner, 2017).Simulation tools were developed in order to capture a large framework of cascading failures in epidemic modeling (Marzo et al., 2017). However, under cascading failures, the system is more robust when the individuals are poorly communicated, in a strong contrast with modern connectivity theory. To the best of our knowledge, there is no simulation tool available for both apparently antipodal scenarios.The graph fragmentation problem (GFP) is the product of a worst-case analysis of a random attack under cascading failures, therefore it is suitable for pandemic analysis. However, in its minmax version, we recover a previous problem called component order connectivity (COC). The corresponding max-min version for COC is a suitable connectivity metric.The goal of this paper is to present a comprehensive analysis for the GFP, its relation with COC, and new feasible vulnerability/connectivity metrics as a synthesis. Both GFP and COC are formally presented in Section 2. Section 3 contains a comprehensive analysis for the GFP. This section covers several approaches for the problem in different subsections, such as Complexity (Subsection 3.1), Approximation algorithms (Subsection 3.2), Polytime methods for special graphs (Subsection 3.3), Exact analysis (Subsection 3.4), and Metaheuristics (Subsection 3.5). Each subsection is enriched with references for further reading. In Section 4, we discuss vulnerability/connectivity metrics suggested by GFP and COC, and a potential interplay with game theory. Finally, Section 5 summarizes the conclusions and trends for fut...

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Section: Discussionmentioning

confidence: 99%

“…A pure greedy notion for the problem is presented in Piccini et al. (), where the protected nodes are iteratively picked minimizing the objective function as large as possible in a step‐by‐step fashion.…”

Section: Discussionmentioning

confidence: 99%

Graph fragmentation problem: analysis and synthesis

Aprile

Castro

Ferreira

et al. 2018

Int Trans Operational Res

Self Cite

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“…The computational complexity of the Graph Fragmentation Problem is here established. In Piccini et al (2016), it is proved that a large set of Node Immunization Problems are at least as hard as the GFP. In Aprile et al (2018), the hardness of the GFP is proved in a more simple way.…”

Section: Computational Complexitymentioning

confidence: 99%

GRASP Heuristics for the Stochastic Weighted Graph Fragmentation Problem

Rosenstock

Piccini

Rela

et al. 2019

Machine Learning, Optimization, and Data Science

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Critical nodes play a major role in network connectivity. Identifying them is important to design efficient strategies to prevent malware or epidemics spread through a network. In this context, the Stochastic Weighted Graph Fragmentation Problem (SWGFP) is a combinatorial optimization problem that belongs to the N P − Complete class. Its objective consists in minimizing the impact of a random attack on a singleton, choosing appropiately a set of nodes to immunize given a restricted budget. In the SWGFP, it is assumed that the attack follows a known probability law and that it affects the whole connected component of the attacked node. In this thesis, a GRASP enriched with Path Relinking algorithm is developed to solve the SWGFP. Its performance is studied under three attack scenarios and compared with a GRASP variant that was previously developed in literature and with a Random heuristic for the problem that picks a set of nodes uniformly at random. Computational experiments show that the algorithm based on Independent Sets which is developed in this thesis, outperforms the other two, with lower expected loss scores and higher robustness.

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