Question 9: (8 points) Given are the following 3 facts:
- The LP-relaxation for the Vertex Cover problem as given in Chapter 1 has
the half integrality property, that means, for every extreme point $x^*$ of the
LP, $x^*_i \in \{0, \frac{1}{2}, 1\}$ for all $i \in V$.
- In general, every LP has at least one optimal solution that is an extreme point
and such a solution can be found in polynomial time.
- Every planar graph is 4-colorable and there exists a polynomial time algorithm
that finds such a 4-coloring.
Give a 3/2-approximation algorithm for Vertex Cover when the input is re-
structed to planar graphs. Hint: Apply LP-rounding and let the rounding depend
on the color.
