An Approximate Solution to the Minimum Vertex Cover Problem: The Hallelujah Algorithm

We present a polynomial-time algorithm for Minimum Vertex Cover achieving an approximation ratio strictly less than 2 for any finite undirected graph with at least one edge, thereby disproving the Unique Games Conjecture. The algorithm reduces the problem to a minimum weighted vertex cover on a degree-1 auxiliary graph using weights \( 1/d_v \), solves it optimally via Cauchy-Schwarz-balanced selection, and projects the solution back to a valid cover. Correctness and the strict sub-2 ratio are rigorously proved. Runtime is \( O(|V|+|E|) \), confirming practical scalability and opening avenues for revisiting UGC-dependent hardness results across combinatorial optimization.