We introduced the use of semidefinite programming for approximation algorithms in Chapter 6. The algorithms of that chapter solve a vector programming relaxation, then choose a random hyperplane (or possibly many hyperplanes) to partition the vectors in some way. The central component of the analysis of these algorithms is Lemma 6.7, which says that the probability of two vectors being separated by a random hyperplane is proportional to the angle between them. In this chapter, we look at ways in which we can broaden both the analysis of algorithms using semidefinite programming, and the algorithms themselves.

To broaden our analytical techniques, we revisit two of the problems we discussed initially in Chapter 6. In particular, we consider the problem of approximating integer quadratic programs, which was introduced in Section 6.3, and the problem of coloring a 3-colorable graph, which was introduced in Section 6.5. In our algorithms in this chapter, we again solve vector programming relaxations of the problems, and choose a random hyperplane by drawing its components from the normal distribution. Here, however, our analysis of the algorithms will rely on several more properties of the normal distribution than we used in the previous chapter; in particular, it will be helpful for us to use bounds on the tail of the normal distribution.