A cornerstone in epidemic modeling is the classical susceptible-infected-removed model, or SIR. In this model, individuals are divided into three classes: susceptible (those who can be infected), infected, and removed (those who suffered the infection and recovered, gaining immunity from further contact with infected individuals). Transitions S → I → R occur at constant rates γ S , γ I . The SIR model is both simple and useful to understand cascading failures in a network. Nevertheless, a shortcoming is the unrealistic assumption of random contacts in a fully mixed large population. More realistic models are available from authoritative literature in the field. They consider a graph and an epidemic spread governed by probabilistic rules. In this paper, a combinatorial optimization problem is introduced using graph-theoretic terminology, inspired by an extremal analysis of epidemic modeling. The contributions are threefold. First, a general node immunization problem is defined for node immunization under budget requirements, using probabilistic networks. The goal is to minimize the expected number of deaths under a particular choice of nodes in the system to be immunized. In the second stage, a highly virulent environment leads to a purely combinatorial problem without probabilistic law, called the graph fragmentation problem (GFP). We prove the corresponding decision version for the GFP belongs to the class of N P-complete problems. As a corollary, SIR-based models also belong to this set. Third, a GRASP (greedy randomized adaptive search procedure) heuristic enriched with a path-relinking post-optimization phase is developed for the GFP. Finally, an experimental analysis is carried out under graphs taken from real-life applications.
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...
A combinatorial optimization problem called Graph Fragmentation Problem (GFP) is introduced. The decision variable is a set of protected nodes, which are deleted from the graph. An attacker picks a non-protected node uniformly at random from the resulting subgraph, and it completely affects the corresponding connected component. The goal is to minimize the expected number of affected nodes S. The GFP finds applications in fire fighting, epidemiology and robust network design among others. A Greedy notion for the GFP is presented. Then, we develop a GRASP heuristic enriched with a Path-Relinking post-optimization phase. Both heuristics are compared on the lights of graphs inspired by a real-world application.
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