Given a \(k\)-coloring of a graph \(G\), show that we can find a vertex cover which is a \(2\bigl(1−\frac{1}{k}\bigr)\) approximation.
Hint: use the \(k\)-coloring on the vertices of \(V_{1/2}\).
In the Point Line Cover problem, we are given a set of \(n\) points in the plane and an integer \(k\), and the goal is to check if there exists a set of \(k\) lines on the plane that contain all the input points.
Show a kernel for this problem with \(\mathcal{O}\left(k^2\right)\) points.