Chapter 19 Approximation via

x In addition, the sirloin which I threw overboard, instead of drifting off into the void, didn't seem to want to leave the rocket and revolved about it, a second artificial satellite, which produced a brief eclipse of the sun every eleven minutes and four seconds. To calm my nerves I calculated till evening the components of its trajectory, as well as the orbital perturbation caused by the presence of the lost wrench. I figured out that for the next six million years the sirloin, rotating about the ship in circular path, would lead the wrench, then catch up with it from behind and pass it again. – The Star Diaries, Stanislaw Lem. In this chapter, we will introduce a powerful technique for " structure " approximation. The basic idea is to perform a search by assigning elements weights, and picking the elements according to their weights. Elements weight indicates their importance. By repeatedly picking elements according to their weights, and updating the weights of objected that are being neglected (i.e., they are more important than the current weights indicate), we end up with a structure that has some desired properties. We will demonstrate this technique for two problems. In the first problem, we will compute a spanning tree of points that has low stabbing number. In the second problem, we will show how the Set Cover problem can be approximated efficiently in geometric settings, yielding a better bound than the general approximation algorithm for this problem. Let P be a set of n points in the plane. We would like to compute a tree T that span the points of P such that every line in the plane crosses the edges of the tree at most O(√ n) times. If the points are the √ n × √ n grid, this easily holds, if we pick any spanning tree that connects only points that are adjacent on the grid. It is not hard to show that one can not do better (see Exercise 19.4.2).