Efficient Algorithms for Total Domination and Total Roman Domination in Unit Disk Graphs

Core Concepts

This paper proposes efficient approximation algorithms for the Total Dominating Set (TDS) and Total Roman Dominating Set (TRDS) problems in unit disk graphs.

Abstract

The paper focuses on the Total Dominating Set (TDS) and Total Roman Dominating Set (TRDS) problems in unit disk graphs (UDGs). Key highlights: The authors prove that the TRDS problem is NP-complete in UDGs. They propose a 7.17-factor approximation algorithm for the TDS problem in UDGs, with a running time of O(n log k), where n is the number of vertices and k is the size of the independent set. They also propose a 6.03-factor approximation algorithm for the TRDS problem in UDGs, with the same time complexity. The algorithms use a greedy set cover approach as a subroutine to efficiently find the total dominating set and total Roman dominating set. The authors provide detailed analysis and proofs to establish the approximation factors and time complexities of the proposed algorithms.

Stats

|V(G)| = n |D| ≤ 44/9 * |D*| |V2| ≤ 44/9 * |D*| W(f) ≤ 2171/360 * W(f*)

Quotes

"The TRDS problem is NP-complete in unit disk graphs (UDGs)." "The proposed algorithm (TDS-UDG-SC) gives a 7.17 -factor approximation result for the TDS problem in UDGs." "The proposed algorithm (TRDF-UDG-SC) gives a 6.03 - factor approximation result for the TRDF problem in UDGs."

Key Insights Distilled From

Improved Total Domination and Total Roman Domination in Unit Disk Graphs

by Sasmita Rout... at arxiv.org 04-05-2024

https://arxiv.org/pdf/2404.03511.pdf

Deeper Inquiries

How can the proposed algorithms be extended to handle dynamic changes in the unit disk graph, such as the addition or removal of vertices and edges

To extend the proposed algorithms to handle dynamic changes in the unit disk graph, such as the addition or removal of vertices and edges, we can implement incremental algorithms. For dynamic changes involving the addition of vertices or edges: Addition of Vertices: When a new vertex is added, we can update the maximal independent set or the set cover instance accordingly. This update can be done efficiently by considering the local neighborhood of the new vertex and its impact on the existing sets. Recalculate the dominating or Roman dominating sets based on the updated information. Addition of Edges: If a new edge is added, we may need to reevaluate the sets to ensure that the total and Roman domination properties are maintained. Update the sets based on the connectivity changes introduced by the new edge. For dynamic changes involving the removal of vertices or edges: Removal of Vertices: If a vertex is removed, we need to adjust the sets by considering the impact of the removal on the domination properties. Recalculate the sets to ensure that the total and Roman domination requirements are still satisfied. Removal of Edges: When an edge is removed, reassess the sets to account for the change in connectivity. Update the sets to reflect the new graph structure without the removed edge. By incorporating these strategies for incremental updates, the algorithms can adapt to dynamic changes in the unit disk graph efficiently.

What are the practical implications of the Total Roman Dominating Set problem in the context of wireless sensor networks or other real-world applications

The Total Roman Dominating Set problem has significant practical implications in wireless sensor networks (WSNs) and other real-world applications. In WSNs: Network Security: Identifying a minimum set of monitoring nodes that collectively ensure total Roman domination can enhance network security. These nodes can detect and respond to security breaches effectively. Fault Tolerance: Total Roman domination ensures that there is always a monitoring node within reach of any part of the network. This aids in fault detection and localization, improving the network's resilience. Resource Optimization: By minimizing the number of monitoring nodes while maintaining total Roman domination, resource utilization in terms of energy and bandwidth can be optimized. Coverage: Total Roman domination guarantees that every node in the network is monitored, enhancing overall coverage and ensuring comprehensive data collection. Overall, the Total Roman Dominating Set problem provides a robust framework for ensuring network integrity, security, and efficiency in wireless sensor networks and similar applications.

Can the techniques used in this paper be applied to other geometric graph problems to obtain efficient approximation algorithms

The techniques used in the paper can be applied to other geometric graph problems to derive efficient approximation algorithms. Some potential applications include: Geometric Spanning Trees: Developing approximation algorithms for minimum spanning trees or Steiner trees in geometric graphs can benefit from similar approaches used in the Total Roman Dominating Set problem. Geometric Clustering: Addressing clustering problems in geometric graphs, such as k-center clustering or facility location problems, using approximation algorithms inspired by the proposed techniques. Geometric Network Design: Optimizing network design in geometric graphs, considering connectivity, coverage, and resource allocation constraints, can leverage the algorithmic strategies employed in the paper. By adapting the methodologies and insights from the Total Roman Dominating Set problem, efficient approximation algorithms can be developed for a wide range of geometric graph problems.