In chapter of Hardness of Approximation in V. Vazirani's book it says:
For establishing a hardness of approximation result for, say, the vertex cover problem, PCP theorem is used to show the following polynomial time reduction. It maps an instance φ of SAT to a graph G = (V,E) such that
• if φ is satisfiable, G has a vertex cover of size ≤ 2/3*|V |, and
• if φ is not satisfiable, the smallest vertex cover in G is of size > α· 2/3*|V |,
Next line it claims:
Claim 29.1 As a consequence of the reduction stated above, there is no polynomial time algorithm for vertex cover that achieves an approximation guarantee of α, assuming P ≠ NP. (α>1)
I couldn't understand that claim, while in previous chapters it says Vertex Cover have 2-approximation algorithm (with Maximal matching).
Could anyone explain what does it means? How it obtained 2/3*|V|? and Why it said there is no α approximation while in previous chapters it proposed ones?