  Treewidth is a useful tool in designing graph algorithms. Although many
NP-hard graph problems can be solved in linear time when the input graphs have
small treewidth, there are problems which remain hard on graphs of bounded
treewidth. In this paper, we consider three vertex selection problems that are
W[1]-hard when parameterized by the treewidth of the input graph, namely the
capacitated vertex cover problem, the target set selection problem and the
vector dominating set problem. We provide two new methods to obtain FPT
approximation algorithms for these problems. For the capacitated vertex cover
problem and the vector dominating set problem, we obtain
$(1+o(1))$-approximation FPT algorithms. For the target set selection problem,
we give an FPT algorithm providing a tradeoff between its running time and the
approximation ratio.
