Vertex Cover, Hardness of Approximation 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 &gt;
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&gt;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?
We can proof this claim by contradiction. If there exists an α-approximation algorithm for Vertex Cover, we will be able to use it for solving SAT problem in polynomial time and it's a contradiction by giving P≠NP.

We assume that there exists a gap introducing reduction that converts an instance of SAT to an instance of Vertex Cover giving:

• if φ is satisfiable, OPT(VC) ≤ 2/3 * |V|, and

• if φ is not satisfiable, OPT(VC) > α * 2/3 * |V|.

We can use this reduction and convert φ to I(VC) in polynomial time, then use the α-approximation algorithm for Vertex Cover in polynomial time and calculate objective function value (we call it f) for it.

As we know value of f stands between two bounds:

OPT(VC) ≤ f ≤ α*OPT(VC)

After that, we can use f to make a decision about φ satisfaction state:

If f ≤ α * 2/3 * |V|, then yes, else no.

We claim that if f is less than α * 2/3 * |V|, φ should be satisfiable because if φ isn't satisfiable, f can't be less than α * 2/3 * |V| which is lower bound to OPT(VC) (according to reduction assumptions) and objective function value can't be less than optimum.

Also we claim that if f is greater than α * 2/3 * |V|, φ shouldn't be satisfiable because if φ is satisfiable, it violates the approximation inequality. (f ≤ α*OPT(VC))