Disproving the Unique Games Conjecture

The Vertex Cover Problem stands as a cornerstone NP-complete challenge in graph theory, requiring identification of a minimal vertex set that covers all edges in an undirected graph $G = (V,E)$. This work introduces the \texttt{find\_vertex\_cover} algorithm, a novel approximation technique that employs polynomial-time reduction to maximum degree-1 instances through auxiliary vertex construction. The method computes approximate solutions via dual optimization approaches---combining weighted dominating sets and vertex covers on reduced graphs---while maintaining an approximation ratio strictly better than the classical 2-approximation bound. With $\mathcal{O}(m\log n)$ complexity for a graph with $n$ vertices and $m$ edges, the algorithm achieves practical efficiency through component-wise processing and linear-space reduction. Implemented in Python, this approach demonstrates particular effectiveness on sparse graphs and scale-free networks, potentially impacting fine-grained complexity conjectures like the Unique Games Conjecture. Practically, it aids applications in network design, scheduling, and bioinformatics by providing near-optimal solutions. This work bridges theoretical advancements and practical utility, offering a promising heuristic for vertex cover approximation.