  We study the problem of deleting the smallest set $S$ of vertices (resp.
edges) from a given graph $G$ such that the induced subgraph (resp. subgraph)
$G \setminus S$ belongs to some class $\mathcal{H}$. We consider the case where
graphs in $\mathcal{H}$ have treewidth bounded by $t$, and give a general
framework to obtain approximation algorithms for both vertex and edge-deletion
settings from approximation algorithms for certain natural graph partitioning
problems called $k$-Subset Vertex Separator and $k$-Subset Edge Separator,
respectively.
  For the vertex deletion setting, our framework combined with the current best
result for $k$-Subset Vertex Separator, yields a significant improvement in the
approximation ratios for basic problems such as $k$-Treewidth Vertex Deletion
and Planar-$F$ Vertex Deletion. Our algorithms are simpler than previous works
and give the first uniform approximation algorithms under the natural
parameterization.
  For the edge deletion setting, we give improved approximation algorithms for
$k$-Subset Edge Separator combining ideas from LP relaxations and important
separators. We present their applications in bounded-degree graphs, and also
give an APX-hardness result for the edge deletion problems.
