# Approximability of Covering Cells with Line Segments

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11346)

## Abstract

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 $$\mathsf {NP}$$-hard, even if the line segments in S are axis-parallel, and it remains $$\mathsf {NP}$$-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 $$\mathsf {PTAS}$$ 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 $$\mathsf {APX}$$-hard. We also consider the parameterized complexity of the problem and prove that the problem is $$\mathsf {FPT}$$ when parameterized by the size of an optimal solution. Our $$\mathsf {FPT}$$ 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.  in which the goal is to cover the “rectangular” cells.

## References

1. 1.
Ashok, P., Kolay, S., Misra, N., Saurabh, S.: Unique covering problems with geometric sets. In: Xu, D., Du, D., Du, D. (eds.) COCOON 2015. LNCS, vol. 9198, pp. 548–558. Springer, Cham (2015).
2. 2.
Bandyapadhyay, S., Maheshwari, A., Mehrabi, S., Suri, S.: Approximating dominating set on intersection graphs of rectangles and L-frames. In: MFCS 2018 (2018, to appear). arXiv version https://arxiv.org/abs/1803.06216
3. 3.
Bose, P., et al.: Coloring and guarding arrangements. Discrete Math. Theor. Comput. Sci. 15(3), 139–154 (2013)
4. 4.
Chan, T.M., Har-Peled, S.: Approximation algorithms for maximum independent set of pseudo-disks. Discrete Comput. Geom. 48(2), 373–392 (2012)
5. 5.
Cormen, T.H., Leiserson, C.E., Rivest, R.L., Stein, C.: Introduction to Algorithms, 2nd edn. The MIT Press, McGraw-Hill Book Company, Cambridge, New York (2001)
6. 6.
Downey, R.G., Fellows, M.R.: Parameterized Complexity. Springer, Heidelberg (1999).
7. 7.
Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity. TCS. Springer, Heidelberg (2013).
8. 8.
Feige, U.: A threshold of ln $$n$$ for approximating set cover. J. ACM 45(4), 634–652 (1998)
9. 9.
Korman, M., Poon, S.-H., Roeloffzen, M.: Line segment covering of cells in arrangements. Inf. Process. Lett. 129, 25–30 (2018)
10. 10.
Mehrabi, S.: Approximating domination on intersection graphs of paths on a grid. In: Solis-Oba, R., Fleischer, R. (eds.) WAOA 2017. LNCS, vol. 10787, pp. 76–89. Springer, Cham (2018).
11. 11.
Mustafa, N.H., Ray, S.: Improved results on geometric hitting set problems. Discrete Comput. Geom. 44(4), 883–895 (2010)
12. 12.
Papadimitriou, C.H., Yannakakis, M.: Optimization, approximation, and complexity classes. J. Comput. Syst. Sci. 43(3), 425–440 (1991)

© Springer Nature Switzerland AG 2018

## Authors and Affiliations

• Paz Carmi
• 1
• Anil Maheshwari
• 2
• Saeed Mehrabi
• 2
Email author
• Luís Fernando Schultz
• 3
• Xavier da Silveira
• 3
1. 1.Department of Computer ScienceBen-Gurion University of the NegevBeer-ShevaIsrael
2. 2.School of Computer ScienceCarleton UniversityOttawaCanada
3. 3.School of Computer Science and Electrical EngineeringUniversity of OttawaOttawaCanada