Graph fragmentation problem: analysis and synthesis

Abstract

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...

GRASP Heuristics for the Stochastic Weighted Graph Fragmentation Problem

“…In that dissertation, the N P-Hardness for the GFP is also established and a first GRASP is developed for the problem. In Natalia Castro's thesis, with external collaboration from Manuel Aprile, it is proved the non-existence of an approximation algorithm with factor lower than 5/3, unless P = N P. Optimal solutions for the GFP are presented for acyclic graphs (Castro (2018); Aprile et al (2017Aprile et al ( , 2018). Additionally, as a result of the doctoral dissertation of Dr. Graciela Ferreira, it was developed a Mixed Integer Linear Programming (MILP) for the GFP and efficient bounds for the optimal value of any instance of the problem (Ferreira (2018)).…”

“…Additionally, as a result of the doctoral dissertation of Dr. Graciela Ferreira, it was developed a Mixed Integer Linear Programming (MILP) for the GFP and efficient bounds for the optimal value of any instance of the problem (Ferreira (2018)). In Piccini et al (2018) a GRASP for the GFP is introduced while in Aprile et al (2018) an analysis and a synthesis of the main up-to-date contributions of the GFP are presented. In this thesis, we will focus on the Stochastic Weighted Graph Fragmentation Problem (SWGFP), a combinatorial optimization problem whose main goal is to find a set of weighted nodes to immunize in order to stop an epidemic spread, once a node is infected.…”

“…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.…”

“…In Aprile et al (2018), it is proved that there is no approximation algorithm for GFP with factor α < 5/3, unless N P = P.…”

Cost-effective Network Disintegration through Targeted Enumeration

“…This so-called network disintegration problem has attracted increasing attention among researchers. [12][13][14][15][16] The core of the network disintegration problem-also known as "network attack," 12,17 "graph fragmentation," 18 and "network dismantling" 15,19 -is determining the node or link set to be removed under certain constraints and various disintegration goals. 20,21 This problem is typically NP-hard for general graphs 22 and its mathematical essence is a combinatorial optimization problem.…”