WADS 2019: Algorithms and Data Structures pp 211-224

# Computing Maximum Independent Set on Outerstring Graphs and Their Relatives

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11646)

## 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 ($$\mathsf {MIS}$$) problem of the underlying graph can be solved 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, $$\mathsf {L}$$-shapes, etc.), then the algorithm takes $$O(n^3)$$ time.

In this paper, we examine the fine-grained complexity of the $$\mathsf {MIS}$$ problem on some well-known outerstring representations. We show that solving the $$\mathsf {MIS}$$ problem on grounded segment and grounded square-$$\mathsf {L}$$ representations is at least as hard as solving $$\mathsf {MIS}$$ on circle graph representations. Note that no $$O(n^{2-\delta })$$-time algorithm, $$\delta >0$$, is known for the $$\mathsf {MIS}$$ 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 $$\mathsf {MIS}$$ 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 $$\mathsf {L}$$-shapes in the plane, we give a $$(4\cdot \log \mathsf {OPT})$$-approximation algorithm for $$\mathsf {MIS}$$ (where $$\mathsf {OPT}$$ denotes the size of an optimal solution), improving the previously best-known $$(4\cdot \log n)$$-approximation algorithm of Biedl and Derka (WADS 2017).

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© Springer Nature Switzerland AG 2019

## Authors and Affiliations

• Prosenjit Bose
• 3
• Paz Carmi
• 1
• Mark J. Keil
• 2
• Anil Maheshwari
• 3
• Saeed Mehrabi
• 3
Email author
• Debajyoti Mondal
• 2
• Michiel Smid
• 3
1. 1.Ben-Gurion University of the NegevBeer-ShevaIsrael