In previous chapters, we have used the iterative relaxation method to obtain approximation algorithms for degree bounded network design problems. In this chapter, we illustrate that similar techniques can be applied to other constrained optimization problems. In the first part, we study the partial vertex cover problem and show an iterative 2-approximation algorithm for the problem. In the second part, we study the multicriteria spanning tree problem and present a polynomial time approximation scheme for the problem.

Vertex cover

We first give a simple iterative 2-approximation algorithm for the vertex cover problem, and then show that it can be extended to the partial vertex cover problem.

Given a graph G = (V, E) and a nonnegative cost function c on vertices, the goal in the vertex cover problem is to find a set of vertices with minimum cost that covers every edge (i.e., for every edge at least one endpoint is in the vertex cover). In Chapter 3, we showed that the vertex cover problem in bipartite graphs is polynomial time solvable and gave an iterative algorithm for finding the minimum cost vertex cover. In general graphs, the vertex cover problem is NP-hard. Nemhauser and Trotter [105] gave a 2-approximation for the problem. Indeed, they prove a stronger property of half-integrality of the natural linear programming relaxation. We prove this result and its extensions to the partial vertex cover problem in the next section.