TY - GEN

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

AU - Bose, Prosenjit

AU - Carmi, Paz

AU - Keil, Mark J.

AU - Maheshwari, Anil

AU - Mehrabi, Saeed

AU - Mondal, Debajyoti

AU - Smid, Michiel

N1 - Publisher Copyright:
© Springer Nature Switzerland AG 2019.

PY - 2019/1/1

Y1 - 2019/1/1

N2 - 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 solved in O(s3) 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(n3) 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(n2-δ) -time algorithm, δ> 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(n2) 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 · 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).

AB - 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 solved in O(s3) 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(n3) 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(n2-δ) -time algorithm, δ> 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(n2) 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 · 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).

UR - http://www.scopus.com/inward/record.url?scp=85070604913&partnerID=8YFLogxK

U2 - 10.1007/978-3-030-24766-9_16

DO - 10.1007/978-3-030-24766-9_16

M3 - Conference contribution

AN - SCOPUS:85070604913

SN - 9783030247652

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 211

EP - 224

BT - Algorithms and Data Structures - 16th International Symposium, WADS 2019, Proceedings

A2 - Friggstad, Zachary

A2 - Salavatipour, Mohammad R.

A2 - Sack, Jörg-Rüdiger

PB - Springer Verlag

T2 - 16th International Symposium on Algorithms and Data Structures, WADS 2019

Y2 - 5 August 2019 through 7 August 2019

ER -