We prove the Minimum Vertex Cover problem to be NP-hard to approximate to within a factor of , extending on previous PCP and hardness of approximation technique. To that end, one needs to develop a new proof framework, and to borrow and extend ideas from several ﬁelds. 1. Introduction The basic purpose of computational complexity theory is to classify computational problems according to the amount of resources required to solve them. In particular, the most basic task is to classify computational problems to those that are eﬃciently solvable and those that are not. .