A 2-Approximation Algorithm for Dominating Sets

The Dominating Set problem, a cornerstone of graph theory and combinatorial optimization, involves finding a subset of vertices in a graph such that every vertex is either in the subset or adjacent to a vertex in it. This problem, known to be NP-hard, has wide-ranging applications, including network design, social network analysis, and resource allocation. In this paper, we introduce an efficient polynomial-time algorithm for computing an approximate Dominating Set in undirected graphs. We rigorously prove that our algorithm achieves a 2-approximation ratio, ensuring that the size of the dominating set it produces is at most twice that of the optimal solution. This result leverages a novel transformation of general graphs into chordal graphs, where a greedy strategy exploits structural properties to deliver the approximation guarantee. Our approach builds on prior work in approximation algorithms, offering a practical solution to a computationally challenging problem. While the Dominating Set problem's NP-hardness suggests that finding an optimal solution in polynomial time is unlikely unless P = NP, our algorithm demonstrates that near-optimal solutions are attainable efficiently. This contribution enhances the toolkit available for tackling NP-hard optimization problems, providing both theoretical insights and practical utility. Future work may explore tightening the approximation ratio or extending the method to weighted graphs, further advancing its applicability in real-world scenarios. MSC Classification: 68Q25, 68R10