TY - JOUR

T1 - Approximability of covering cells with line segments

AU - Carmi, Paz

AU - Maheshwari, Anil

AU - Mehrabi, Saeed

AU - Xavier da Silveira, Luís Fernando Schultz

N1 - Funding Information:
The research of Paz Carmi is supported by Grant 2016116 from the United States-Israel Binational Science Foundation . The research of Anil Maheshwari and Saeed Mehrabi is supported in part by Natural Sciences and Engineering Research Council of Canada (NSERC). We thank the anonymous reviewers for their helpful comments, and for pointing us out to [23] .
Publisher Copyright:
© 2019

PY - 2019/9/13

Y1 - 2019/9/13

N2 - Korman et al. [18] 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 to 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. [18] in which the goal is to cover the “rectangular” cells.

AB - Korman et al. [18] 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 to 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. [18] in which the goal is to cover the “rectangular” cells.

KW - APX-hardness

KW - Approximation algorithms

KW - Fixed-parameter tractability

KW - Geometric covering

KW - Line segment covering

KW - PTAS

UR - http://www.scopus.com/inward/record.url?scp=85066084807&partnerID=8YFLogxK

U2 - 10.1016/j.tcs.2019.05.004

DO - 10.1016/j.tcs.2019.05.004

M3 - Article

AN - SCOPUS:85066084807

VL - 784

SP - 133

EP - 141

JO - Theoretical Computer Science

JF - Theoretical Computer Science

SN - 0304-3975

ER -