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Computing Maximum Independent Set on Outerstring Graphs and Their Relatives

  • Prosenjit Bose
  • Paz Carmi
  • Mark J. Keil
  • Anil Maheshwari
  • Saeed MehrabiEmail author
  • Debajyoti Mondal
  • Michiel Smid
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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Copyright information

© 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
  2. 2.University of SaskatchewanSaskatoonCanada
  3. 3.Carleton UniversityOttawaCanada

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