Algorithms for some graph theoretical optimization problems

This thesis is situated in the field of combinatorial optimization. More specifically, we focus on solution methods for solving a number of graphtheoretical optimization problems. First we give a short introduction to the field of integer programming and state of the art solution techniques. The remainder of this thesis can be roughly split into two parts. The first part is dedicated to the problem of partitioning partially ordered sets. The second part deals with the structure and the connectivity of the Internet. The problem of partitioning a partially ordered set into a minimal number of chains, such that each element belongs to at least one chain, is a basic and fundamental problem in operations research. Dilworth (1950) showed that it is solvable in polynomial time, and that the minimum number of chains needed to cover all elements of X is equal to the value of a maximum antichain. We generalize this problem by assuming that the chains must have bounded size, and we propose a number of exact algorithms for solving this problem. We apply these algorithms to a real-world application of this problem encountered at a manufacturing company in the Netherlands. One of the interesting outcomes of this work is that, in this real-world setting, we were able to identify a special structure in the problem instances. It turns out that these problem instances have a property called bounded clique-width, which allows us to design a polynomial time algorithm for these special instances that works extremely well.