  Let $\mathcal{C}$ and $\mathcal{D}$ be hereditary graph classes. Consider the
following problem: given a graph $G\in\mathcal{D}$, find a largest, in terms of
the number of vertices, induced subgraph of $G$ that belongs to $\mathcal{C}$.
We prove that it can be solved in $2^{o(n)}$ time, where $n$ is the number of
vertices of $G$, if the following conditions are satisfied:
  * the graphs in $\mathcal{C}$ are sparse, i.e., they have linearly many edges
in terms of the number of vertices;
  * the graphs in $\mathcal{D}$ admit balanced separators of size governed by
their density, e.g., $\mathcal{O}(\Delta)$ or $\mathcal{O}(\sqrt{m})$, where
$\Delta$ and $m$ denote the maximum degree and the number of edges,
respectively; and
  * the considered problem admits a single-exponential fixed-parameter
algorithm when parameterized by the treewidth of the input graph.
  This leads, for example, to the following corollaries for specific classes
$\mathcal{C}$ and $\mathcal{D}$:
  * a largest induced forest in a $P_t$-free graph can be found in
$2^{\tilde{\mathcal{O}}(n^{2/3})}$ time, for every fixed $t$; and
  * a largest induced planar graph in a string graph can be found in
$2^{\tilde{\mathcal{O}}(n^{3/4})}$ time.
