Computing maximum independent set on outerstring graphs and their relatives

Abstract

A graph G with n vertices is called an outerstring graph if it has an intersection representation with 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 of G with s segments, a Maximum Independent Set (MIS) of G can be computed in O(s3) time (Keil et al. (2017) [22]).We examine the fine-grained complexity of the MIS problem on some well-known outerstring representations (e.g., line segments, L-shapes, etc.), where the strings are of constant size. We show that computing MIS on grounded segment and grounded square-L representations is at least as hard as computing MIS on circle graph representations. Note that no O(n2−δ)-time algorithm, δ>0, is known for computing MIS 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(n2) time and show this to be the best possible under the Strong Exponential Time Hypothesis. For the intersection graph of nL-shapes in the plane, we give a (4⋅log⁡OPT)-approximation algorithm for MIS (where OPT denotes the size of an optimal solution), improving the previously best-known (4⋅log⁡n)-approximation algorithm of Biedl and Derka (WADS 2017).