  For a family of graphs $\cal F$, the canonical Weighted $\cal F$ Vertex
Deletion problem is defined as follows: given an $n$-vertex undirected graph
$G$ and a weight function $w: V(G)\rightarrow\mathbb{R}$, find a minimum weight
subset $S\subseteq V(G)$ such that $G-S$ belongs to $\cal F$. We devise a
recursive scheme to obtain $O(\log^{O(1)}n)$-approximation algorithms for such
problems, building upon the classic technique of finding balanced separators in
a graph. Roughly speaking, our scheme applies to problems where an optimum
solution $S$, together with a well-structured set $X$, form a balanced
separator of $G$. We obtain the first $O(\log^{O(1)}n)$-approximation
algorithms for the following problems.
  * We give an $O(\log^2n)$-factor approximation algorithm for Weighted Chordal
Vertex Deletion (WCVD), the vertex deletion problem to the family of chordal
graphs. On the way, we also obtain a constant factor approximation algorithm
for Multicut on chordal graphs.
  * We give an $O(\log^3n)$-factor approximation algorithm for Weighted
Distance Hereditary Vertex Deletion (WDHVD). This is the vertex deletion
problem to the family of distance hereditary graphs, or equivalently, the
family of graphs of rankwidth 1.
  Our methods also allow us to obtain in a clean fashion a
$O(\log^{1.5}n)$-approximation algorithm for the Weighted $\cal F$ Vertex
Deletion problem when $\cal F$ is a minor closed family excluding at least one
planar graph. For the unweighted version of the problem constant factor
approximation algorithms are were known~[Fomin et al., FOCS~2012], while for
the weighted version considered here an $O(\log n \log\log n)$-approximation
algorithm follows from~[Bansal et al., SODA~2017]. We believe that our
recursive scheme can be applied to obtain $O(\log^{O(1)}n)$-approximation
algorithms for many other problems as well.
