  An outerstring graph is the intersection graph of curves lying inside a disk
with one endpoint on the boundary of the disk. We show that an outerstring
graph with $n$ vertices has treewidth $O(\alpha\log n)$, where $\alpha$ denotes
the arboricity of the graph, with an almost matching lower bound of
$\Omega(\alpha \log (n/\alpha))$. As a corollary, we show that a
$t$-biclique-free outerstring graph has treewidth $O(t(\log t)\log n)$. This
leads to polynomial-time algorithms for most of the central NP-complete
problems such as \textsc{Independent Set}, \textsc{Vertex Cover},
\textsc{Dominating Set}, \textsc{Feedback Vertex Set}, \textsc{Coloring} for
sparse outerstring graphs. Also, we can obtain subexponential-time (exact,
parameterized, and approximation) algorithms for various NP-complete problems
such as \textsc{Vertex Cover}, \textsc{Feedback Vertex Set} and \textsc{Cycle
Packing} for (not necessarily sparse) outerstring graphs.
