3.1 Linear Programming Duality

In the previous few lectures we have seen examples of LP-rounding, a method for obtaining approximation algorithms that involves solving a linear programming relaxation of the problem at hand and rounding the solution. In the last lecture we also discussed the basic theory of LP-duality. Today we will apply this theory to obtain a second LP-based technique for obtaining approximation algorithms — the primal-dual method. This technique has the advantage that it circumvents the need to actually solve an LP relaxation, leading to efficient algorithms that are purely combinatorial. We will apply this technique to the Vertex Cover and Steiner Forest problems. The primal-dual method in the context of approximation algorithms was first used by Goemans and Williamson [1].