Parameterized Complexity of Immunization in the Threshold Model

TL;DR

This paper investigates the computational complexity of controlling contagion spread in networks modeled by the linear threshold model, focusing on parameterized complexity and fixed-parameter algorithms for immunization strategies.

Contribution

It provides a detailed parameterized complexity analysis of immunization problems in the linear threshold model, including hardness results and fixed-parameter algorithms for various network parameters.

Findings

Several parameters lead to W[1] or W[2]-hardness results.

Fixed-parameter algorithms are developed for certain parameter combinations.

The study advances understanding of the computational limits of immunization strategies in network diffusion models.

Abstract

We consider the problem of controlling 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 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 $\ell$, is it possible to limit the diffusion to at most $k$ other nodes of the network by immunizing at most $\ell$ nodes? We consider several parameters associated to the input, including: the bounds $k$ and $\ell$, the maximum node degree $\Delta$, 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.