Example

Optimal = {g, h}

Approximate = {a, g, d, h}

The size of the approximate vertex cover is never more than twice the size of the optimal vertex cover.

Theorem 37.1

Approx-Vertex-Cover has a ratio bound of 2.

Proof:

The approximate solution C is a vertex cover.

Let A be the set of edges chosen by the algorithm. Since each such edge's endpoints were not in C at the time, $\mid$C$\mid$ = 2$\mid$A$\mid$. An optimal cover must have at least $\mid$A$\mid$ vertices, $\mid$C*$\mid$$\geq$$\mid$A$\mid$. Thus $\mid$C*$\mid$$\geq$ 1/2$\mid$C$\mid$ and $\leq$ 2 = p.