  A graph $G$ with $n$ vertices is called an outerstring graph if it has an
intersection representation of a set of $n$ curves inside a disk such that one
endpoint of every curve is attached to the boundary of the disk. Given an
outerstring graph representation, the Maximum Independent Set (MIS) problem of
the underlying graph can be computed in $O(s^3)$ time, where $s$ is the number
of segments in the representation (Keil et al., Comput. Geom., 60:19--25,
2017). If the strings are of constant size (e.g., line segments, L-shapes,
etc.), then the algorithm takes $O(n^3)$ time.
  In this paper, we examine the fine-grained complexity of the MIS problem on
some well-known outerstring representations. We show that solving the MIS
problem on grounded segment and grounded square-L representations is at least
as hard as solving MIS on circle graph representations. Note that no
$O(n^{2-\delta})$-time algorithm, $\delta>0$, is known for the MIS problem on
circle graphs. For the grounded string representations where the strings are
$y$-monotone simple polygonal paths of constant length with segments at
integral coordinates, we solve MIS in $O(n^2)$ time and show this to be the
best possible under the strong exponential time hypothesis (SETH). For the
intersection graph of $n$ L-shapes in the plane, we give a $(4\cdot \log
OPT)$-approximation algorithm for MIS (where $OPT$ denotes the size of an
optimal solution), improving the previously best-known $(4\cdot \log
n)$-approximation algorithm of Biedl and Derka (WADS 2017).
