TY - GEN

T1 - Approximability of covering cells with line segments

AU - Carmi, Paz

AU - Maheshwari, Anil

AU - Mehrabi, Saeed

AU - Schultz, Luís Fernando

AU - da Silveira, Xavier

N1 - Funding Information: Paz Carmi is supported by Grant 2016116 from the United States-Israel Binational Science Foundation. Anil Maheshwari is supported in part by Natural Sciences and Engineering Research Council of Canada (NSERC). Saeed Mehrabi is supported by a Carleton-Fields postdoctoral fellowship. Publisher Copyright: © 2018, Springer Nature Switzerland AG.

PY - 2018/1/1

Y1 - 2018/1/1

N2 - In COCOA 2015, Korman et al. studied the following geometric covering problem: given a set S of n line segments in the plane, find a minimum number of line segments such that every cell in the arrangement of the line segments is covered. Here, a line segment s covers a cell f if s is incident to f. The problem was shown to be NP-hard, even if the line segments in S are axis-parallel, and it remains NP-hard when the goal is cover the "rectangular" cells (i.e., cells that are defined by exactly four axis-parallel line segments). In this paper, we consider the approximability of the problem. We first give a PTAS for the problem when the line segments in S are in any orientation, but we can only select the covering line segments from one orientation. Then, we show that when the goal is to cover the rectangular cells using line segments from both horizontal and vertical line segments, then the problem is APX-hard. We also consider the parameterized complexity of the problem and prove that the problem is FPT when parameterized by the size of an optimal solution. Our FPT algorithm works when the line segments in S have two orientations and the goal is to cover all cells, complementing that of Korman et al. [9] in which the goal is to cover the "rectangular" cells.

AB - In COCOA 2015, Korman et al. studied the following geometric covering problem: given a set S of n line segments in the plane, find a minimum number of line segments such that every cell in the arrangement of the line segments is covered. Here, a line segment s covers a cell f if s is incident to f. The problem was shown to be NP-hard, even if the line segments in S are axis-parallel, and it remains NP-hard when the goal is cover the "rectangular" cells (i.e., cells that are defined by exactly four axis-parallel line segments). In this paper, we consider the approximability of the problem. We first give a PTAS for the problem when the line segments in S are in any orientation, but we can only select the covering line segments from one orientation. Then, we show that when the goal is to cover the rectangular cells using line segments from both horizontal and vertical line segments, then the problem is APX-hard. We also consider the parameterized complexity of the problem and prove that the problem is FPT when parameterized by the size of an optimal solution. Our FPT algorithm works when the line segments in S have two orientations and the goal is to cover all cells, complementing that of Korman et al. [9] in which the goal is to cover the "rectangular" cells.

UR - http://www.scopus.com/inward/record.url?scp=85058539855&partnerID=8YFLogxK

U2 - https://doi.org/10.1007/978-3-030-04651-4_29

DO - https://doi.org/10.1007/978-3-030-04651-4_29

M3 - Conference contribution

SN - 9783030046507

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

SP - 436

EP - 448

BT - Combinatorial Optimization and Applications - 12th International Conference, COCOA 2018, Proceedings

A2 - Zelikovsky, Alexander

A2 - Kim, Donghyun

A2 - Uma, R.N.

PB - Springer Verlag

T2 - 12th Annual International Conference on Combinatorial Optimization and Applications, COCOA 2018

Y2 - 15 December 2018 through 17 December 2018

ER -