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Example



\psfig{figure=figures/f28-1.ps}

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 35.1

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 \(\frac{\mid C \mid}{\mid C* \mid}\)$\leq$ 2 = p.


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