Exact Algorithms for Generalizations of Vertex Cover

The NP-complete Vertex Cover problem has been intensively studied in the field of parameterized complexity theory. However, there exists only little work concerning important generalizations of Vertex Cover like Partial Vertex Cover, Connected Vertex Cover, and Capacitated Vertex Cover which are of high interest in theory as well as in real-world applications. So far research was mainly focused on the approximability of these problems. It was shown recently that, with the size of the vertex cover as parameter, Connected Vertex Cover and Capacitated Vertex Cover are both fixed-parameter tractable whereas Partial Vertex Cover is W [1]-hard. We will study the fixed-parameter tractability of these problems using another parameter, called the treewidth, which describes the “tree-likeness” of the input graph. Our dynamic programming approaches lead to exact algorithms for graph classes with small treewidth. With these algorithms we show that Partial Vertex Cover and Connected Vertex Cover are fixed-parameter tractable using treewidth as a parameter, and that Capacitated Vertex Cover is fixed-parameter tractable with respect to treewidth for graphs with bounded vertex degree. Additionally, we will consider a variant of Capacitated Vertex Cover which is NPcomplete for trees. For this problem we show that it is fixed-parameter tractable when parameterized by the vertex degree.