Assigning times to minimise reachability in temporal graphs

TL;DR

This paper explores how to order active times of edges in temporal graphs to minimize the maximum reachability, which models worst-case disease spread, and shows the problem is NP-hard with some solvable cases.

Contribution

It introduces a novel modification approach for temporal graphs by fixing active times and changing their order, and analyzes the computational complexity of the resulting optimization problem.

Findings

The problem of ordering edges to minimize reachability is NP-hard.

Certain cases of the problem can be solved efficiently.

Approximation algorithms are identified for some instances.

Abstract

Temporal graphs (in which edges are active at specified times) are of particular relevance for spreading processes on graphs, e.g.~the spread of disease or dissemination of information. Motivated by real-world applications, modification of static graphs to control this spread has proven a rich topic for previous research. Here, we introduce a new type of modification for temporal graphs: the number of active times for each edge is fixed, but we can change the relative order in which (sets of) edges are active. We investigate the problem of determining an ordering of edges that minimises the maximum number of vertices reachable from any single starting vertex; epidemiologically, this corresponds to the worst-case number of vertices infected in a single disease outbreak. We study two versions of this problem, both of which we show to be $\NP$-hard, and identify cases in which the problem can be solved or approximated efficiently.