# Computing Maximum Independent Set on Outerstring Graphs and Their Relatives

## TL;DR

This paper investigates the computational complexity of the Maximum Independent Set problem on various classes of outerstring graphs, providing new algorithms, complexity bounds, and approximation results for these geometric intersection graphs.

## Contribution

It establishes the hardness of MIS on grounded segment and square-L representations, provides an optimal quadratic-time algorithm for certain string representations under SETH, and improves approximation algorithms for L-shape intersection graphs.

## Key findings

- MIS on grounded segment and square-L representations is as hard as on circle graphs.
- An $O(n^2)$-time algorithm for MIS on y-monotone polygonal paths with constant length.
- A $(4 imes ext{log } OPT)$-approximation for MIS on L-shapes, improving previous bounds.

## Abstract

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).