Benders decomposition algorithms for minimizing the spread of harmful contagions in networks

TL;DR

The paper tackles minimizing the spread of harmful contagions on networks by blocking arc-labels under a budget, using the independent cascade diffusion model. It introduces the measure-based spread minimization problem (MBSMP) and develops exact solution approaches based on Branch-and-Benders-cut for two mixed-integer formulations (arc-based ABF and path-based PBF), with the striking result that Benders optimality cuts can be separated combinatorially via shortest-path computations rather than linear programming subproblems. It strengthens the framework with enhancements such as scenario-dependent extended seed sets, cut sampling, initial cuts, and a starting heuristic, and demonstrates substantial performance gains on real and synthetic networks. The findings show the proposed BC algorithms scale well to large networks, with lifting and fractional separation offering notable improvements, and indicate robustness to variations in diffusion parameters. The work advances exact, scalable strategies for diffusion interdiction problems and suggests several avenues for extension, including node-based blocking, partial blocking, and alternative decomposition schemes.

Abstract

The COVID-19 pandemic has been a recent example for the spread of a harmful contagion in large populations. Moreover, the spread of harmful contagions is not only restricted to an infectious disease, but is also relevant to computer viruses and malware in computer networks. Furthermore, the spread of fake news and propaganda in online social networks is also of major concern. In this study, we introduce the measure-based spread minimization problem (MBSMP), which can help policy makers in minimizing the spread of harmful contagions in large networks. We develop exact solution methods based on branch-and-Benders-cut algorithms that make use of the application of Benders decomposition method to two different mixed-integer programming formulations of the MBSMP: an arc-based formulation and a path-based formulation. We show that for both formulations the Benders optimality cuts can be generated using a combinatorial procedure rather than solving the dual subproblems using linear programming. Additional improvements such as using scenario-dependent extended seed sets, initial cuts, and a starting heuristic are also incorporated into our branch-and-Benders-cut algorithms. We investigate the contribution of various components of the solution algorithms to the performance on the basis of computational results obtained on a set of instances derived from existing ones in the literature.

Figures (9)

• Figure 1: An instance of MBSMP with six nodes and four arc labels, seed set$I=\{1,4\}$ (shown in light grey) and $|\Omega|=3$. For the scenarios, the live-arcs are shown in \ref{['fig:example_s1']}, \ref{['fig:example_s2']} and \ref{['fig:example_s3']}. The budget allows to block at most one label. The solution $K'=\{2\}$ is illustrated, with the blocked arcs shown as dashed arrows. The nodes reached by the spread in each scenario (excluding the seed set) are displayed in dark grey.
• Figure 2: The final spread on HepPh collaboration network with 12,008 nodes and 118,521 edges, (a) when no labels are blocked and (b) when four labels are blocked. There are 50 seed nodes, 50 diffusion scenarios and 21 labels. The filling of the nodes depends on the number of scenarios each node is activated in. The darkest blue filling corresponds to the seed nodes, i.e., nodes which are activated in each of 50 scenarios. Arcs with blocked labels are removed from the graph in (b).
• Figure 3: A small directed network with four labels.
• Figure 4: Reachable nodes and activation paths of a scenario for a given blocking decision
• Figure 5: Degree distributions of the instances

Theorems & Definitions (24)

• Definition 1: Measure-based spread minimization problem (MBSMP)
• Remark 1
• Proposition 1
• proof
• Proposition 2
• proof
• Definition 2
• Definition 3
• Example 1
• Theorem 1