Computational Complexity of Temporal Reachability Dominating Sets in Temporal Graphs

Concepts de base

Determining the minimum number of initially infected individuals required to infect the entire population in a dynamic network is a computationally hard problem, with hardness results depending on the specific temporal constraints.

Résumé

The paper studies the Temporal Reachability Dominating Set (TaRDiS) problem on temporal graphs, which asks whether there exists a small set of k initially infected individuals that can indirectly infect the entire population. The authors provide a comprehensive analysis of the computational complexity of TaRDiS and its variants, considering different temporal constraints.

Key highlights:

• TaRDiS is shown to be NP-complete, even when the lifetime of the temporal graph is bounded or the footprint graph is planar.
• The authors identify the maximum lifetime τ for which each variant of TaRDiS is tractable, demonstrating that the problem becomes intractable for larger lifetimes.
• Parameterized complexity results are provided, showing that TaRDiS is fixed-parameter tractable with respect to the treewidth of the footprint graph and the lifetime.
• The authors also introduce and study the MaxMinTaRDiS problem, which aims to schedule connections between individuals to maximize the minimum size of any TaRDiS. They show this problem to be computationally hard in various settings.
• Interestingly, the authors establish a connection between Nonstrict MaxMinTaRDiS and the well-studied Distance-3 Independent Set problem.

Stats

The paper does not contain any explicit numerical data or statistics to support the key arguments.

Citations

"Given a population with dynamic pairwise connections, we ask if the entire population could be (indirectly) infected by a small group of k initially infected individuals."
"We formalise this problem as the Temporal Reachability Dominating Set (TaRDiS) problem on temporal graphs."
"We show these to be coNP-complete, NP-hard, and ΣP2-complete, respectively."

Idées clés tirées de

Temporal Reachability Dominating Sets: contagion in temporal graphs

by David C. Kut... à arxiv.org 05-06-2024

https://arxiv.org/pdf/2306.06999.pdf

Questions plus approfondies

How can the techniques and insights from this work be applied to study the computational complexity of other temporal graph problems beyond reachability and domination

The techniques and insights from this work can be applied to study the computational complexity of other temporal graph problems by considering similar problems that involve reachability and domination in dynamic networks. For example, problems related to information spreading, influence maximization, and epidemic control in temporal networks can benefit from the analysis of reachability dominating sets. By adapting the framework developed in this paper, researchers can investigate the complexity of problems such as optimal information dissemination, targeted vaccination strategies, and efficient resource allocation in evolving networks. Additionally, the parameterized complexity results obtained in this work can serve as a foundation for analyzing the computational hardness of various temporal graph problems with different parameters and constraints.

What are the practical implications of the hardness results presented in this paper, and how can they inform the design of efficient algorithms for real-world temporal network applications

The hardness results presented in this paper have significant practical implications for the design and implementation of algorithms for real-world temporal network applications. Understanding the computational complexity of problems like TaRDiS and MaxMinTaRDiS can guide researchers and practitioners in developing efficient algorithms for tasks such as identifying influential nodes in social networks, modeling the spread of diseases in populations, and optimizing communication strategies in dynamic environments. By leveraging the insights from the hardness results, algorithm designers can focus on developing approximation algorithms, heuristic approaches, or specialized techniques to tackle these challenging problems effectively. Moreover, the results can inform the development of tools and software for analyzing and managing temporal networks, providing valuable insights for decision-making and strategic planning in various domains.

Are there any natural restrictions or relaxations of the TaRDiS and MaxMinTaRDiS problems that could lead to tractable variants while still capturing important aspects of temporal network dynamics

There are several natural restrictions and relaxations of the TaRDiS and MaxMinTaRDiS problems that could lead to tractable variants while still capturing important aspects of temporal network dynamics. One possible relaxation could involve considering a probabilistic or stochastic version of the problems, where the goal is to find a set of nodes that maximizes the probability of reaching all other nodes in the network over time. This probabilistic approach could lead to efficient algorithms based on Monte Carlo simulations or Markov chain models. Another potential relaxation could be to introduce constraints on the types of interactions or edges allowed in the temporal graph, such as limiting the number of simultaneous connections or imposing restrictions on edge activation patterns. By exploring these variations and incorporating domain-specific knowledge, researchers can develop specialized algorithms that strike a balance between computational complexity and practical applicability in temporal network analysis.