Algorithm Achieves 2-Approximation for Dominating Sets in General Graphs via Bipartite Transformation

This article presents a novel algorithm for approximating minimum dominating sets in general undirected graphs, achieving a 2-approximation ratio through bipartite graph transformation. The algorithm first isolates and handles nodes with no edges, then constructs a bipartite graph where each original node is split into two mirrored vertices. A greedy approach is applied to this bipartite structure to select vertices that maximize coverage of undominated nodes. Theoretical analysis confirms the 2-approximation guarantee by associating each optimal solution vertex with at most two selected vertices. Runtime complexity is O(n² log n) due to sorting operations, with experimental validation on DIMACS benchmark instances demonstrating competitive performance against NetworkX’s logarithmic-approximation baseline. The method provides a deterministic alternative to existing approximation techniques while maintaining polynomial-time efficiency.