TY - JOUR

T1 - Computing maximum independent set on outerstring graphs and their relatives

AU - Bose, Prosenjit

AU - Carmi, Paz

AU - Keil, J. Mark

AU - Maheshwari, Anil

AU - Mehrabi, Saeed

AU - Mondal, Debajyoti

AU - Smid, Michiel

N1 - Funding Information:
A preliminary version of this paper appeared in the 16th International Symposium on Algorithms and Data Structures (WADS 2019) [7]. Research of Prosenjit Bose, Anil Maheshwari, Debajyoti Mondal and Michiel Smid is supported in part by the Natural Sciences and Engineering Research Council of Canada (NSERC). Part of this work was done when Saeed Mehrabi was visiting the University of Saskatchewan.
Funding Information:
A preliminary version of this paper appeared in the 16th International Symposium on Algorithms and Data Structures (WADS 2019) [7] . Research of Prosenjit Bose, Anil Maheshwari, Debajyoti Mondal and Michiel Smid is supported in part by the Natural Sciences and Engineering Research Council of Canada (NSERC). Part of this work was done when Saeed Mehrabi was visiting the University of Saskatchewan.
Publisher Copyright:
© 2021 Elsevier B.V.

PY - 2022/4/1

Y1 - 2022/4/1

N2 - 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 n L-shapes in the plane, we give a (4⋅logOPT)-approximation algorithm for MIS (where OPT denotes the size of an optimal solution), improving the previously best-known (4⋅logn)-approximation algorithm of Biedl and Derka (WADS 2017).

AB - 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 n L-shapes in the plane, we give a (4⋅logOPT)-approximation algorithm for MIS (where OPT denotes the size of an optimal solution), improving the previously best-known (4⋅logn)-approximation algorithm of Biedl and Derka (WADS 2017).

KW - Circle graphs

KW - Fine-grained complexity

KW - Maximum independent set problem

KW - Outerstring graphs

UR - http://www.scopus.com/inward/record.url?scp=85120173836&partnerID=8YFLogxK

U2 - 10.1016/j.comgeo.2021.101852

DO - 10.1016/j.comgeo.2021.101852

M3 - Article

AN - SCOPUS:85120173836

VL - 103

JO - Computational Geometry: Theory and Applications

JF - Computational Geometry: Theory and Applications

SN - 0925-7721

M1 - 101852

ER -