The Minimum Vertex Cover Problem, a cornerstone NP-hard challenge in combinatorial optimization, finds critical applications in network security, bioinformatics, and scheduling. We present a novel polynomial-time approximation algorithm achieving an approximation ratio strictly below $\sqrt{2}$. Our method strategically combines maximum spanning tree properties with bipartite graph decomposition to iteratively construct vertex covers through a dual-stage optimization process. Theoretical analysis confirms the robustness and efficiency of our approach, particularly in handling sparse and large-scale graphs. The polynomial time solution was implemented in Python and it is available at https://pypi.org/project/capablanca/
The Minimum Vertex Cover Problem, a cornerstone NP-hard challenge in combinatorial optimization, finds critical applications in network security, bioinformatics, and scheduling. We present a novel polynomial-time approximation algorithm achieving an approximation ratio strictly below $\sqrt{2}$. Our method strategically combines maximum spanning tree properties with bipartite graph decomposition to iteratively construct vertex covers through a dual-stage optimization process. Theoretical analysis confirms the robustness and efficiency of our approach, particularly in handling sparse and large-scale graphs. The polynomial time solution was implemented in Python and it is available at https://pypi.org/project/capablanca/