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).

翻译：如果磁盘内有一组美元曲线的交叉表示值,则将每个曲线的一个端点附加到磁盘的边界上。鉴于外线图形的表示值,下方图形的最大独立设置(MIS)问题可以以美元计算,其中美元是代表区块数(Keil et al.,Comput. Geom., 60:19-25, 2017)。如果字符串是固定的(例如,线段, L-shapes,等等),那么算法将花费O(n)3美元的时间连接到磁盘的边界上。在本文中,我们检查某些著名的外线表示值显示MIS问题的细微复杂度。我们显示,在基段和平方表示区块上解决MIS问题至少是困难的,在圆形图形表上(n2-delta),在时间平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面平面。注意,在平面平面平面平面平面平面平面平平平平面平面平面平面平面平面平面平面平面平面平面平平平平面平面平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平平面平面平平平平平平平平平平平面。