On the Complexity of Target Set Selection in Simple Geometric Networks

TL;DR

This paper investigates the computational complexity of selecting initial infected individuals to spread a disease in geometric social networks, revealing hardness results and exploring algorithms for simpler graph classes.

Contribution

It proves the problem's hardness on unit disk graphs and provides algorithmic insights for interval and grid graphs.

Findings

Finding minimal initial infected sets is NP-hard on unit disk graphs.

Efficient algorithms are developed for interval and grid graph classes.

The study bridges disease spread modeling with geometric graph theory.

Abstract

We study the following model of disease spread in a social network. At first, all individuals are either infected or healthy. Next, in discrete rounds, the disease spreads in the network from infected to healthy individuals such that a healthy individual gets infected if and only if a sufficient number of its direct neighbors are already infected. We represent the social network as a graph. Inspired by the real-world restrictions in the current epidemic, especially by social and physical distancing requirements, we restrict ourselves to networks that can be represented as geometric intersection graphs. We show that finding a minimal vertex set of initially infected individuals to spread the disease in the whole network is computationally hard, already on unit disk graphs. Hence, to provide some algorithmic results, we focus ourselves on simpler geometric graph classes, such as interval graphs and grid graphs.