Abstract
We study several variations of line segment covering problem with axis-parallel unit squares in the plane. Given a set S of n line segments, the objective is to find the minimum number of axis-parallel unit squares which cover at least one end-point of each segment. The variations depend on the orientation and length of the input segments. We prove some of these problems to be NP-complete, and give constant factor approximation algorithms for those problems. For the general version of the problem, where the segments are of arbitrary length and orientation, and the squares are given as input, we propose a factor 16 approximation result based on multilevel linear programming relaxation technique. This technique may be of independent interest for solving some other problems. We also show that our problems have connections with the problems studied by Arkin et al. [2] on conflict-free covering problem. Our NP-completeness results hold for more simplified types of objects than those of Arkin et al. [2].
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Arkin, E.M., Banik, A., Carmi, P., Citovsky, G., Katz, M.J., Mitchell, J.S.B., Simakov, M.: Choice is hard. In: Elbassioni, K., Makino, K. (eds.) ISAAC 2015. LNCS, vol. 9472, pp. 318–328. Springer, Heidelberg (2015). doi:10.1007/978-3-662-48971-0_28
Arkin, E.M., Banik, A., Carmi, P., Citovsky, G., Katz, M.J., Mitchell, J.S.B., Simakov, M.: Conflict-free covering. In: CCCG (2015)
Bansal, N., Pruhs, K.: The geometry of scheduling. SIAM J. Comput. 43(5), 1684–1698 (2014)
Biniaz, A., Liu, P., Maheshwari, A., Smid, M.: Approximation algorithms for the unit disk cover problem in 2D and 3D. Computational Geometry 60, 8–18 (2016)
Chan, T.M., Grant, E., Konemann, J., Sharpe, M.: Weighted capacited, priority, and geometric set cover via improved quasi-uniform sampling. In: SODA, pp. 1576–1585 (2012)
Gaur, D.R., Ibaraki, T., Krishnamurti, R.: Constant ratio approximation algorithms for the rectangle stabbing problem and the rectilinear partitioning problem. Journal of Algorithms 43(1), 138–152 (2002)
Hochbaum, D.S., Maass, W.: Approximation schemes for covering and packing problems in image processing and VLSI. J. ACM 32(1), 130–136 (1985)
Knuth, D.E., Raghunathan, A.: The problem of compatible representatives. SIAM Journal on Discrete Mathematics 5(3), 422–427 (1992)
Kobylkin, K.: Computational complexity of guarding of proximity graphs. CoRR, abs/1605.00313, 2016
Madireddy, R.R., Mudgal, A.: Stabbing line segments with disks and related problems. In: CCCG (2016)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2017 Springer International Publishing AG
About this paper
Cite this paper
Acharyya, A., Nandy, S.C., Pandit, S., Roy, S. (2017). Covering Segments with Unit Squares. In: Ellen, F., Kolokolova, A., Sack, JR. (eds) Algorithms and Data Structures. WADS 2017. Lecture Notes in Computer Science(), vol 10389. Springer, Cham. https://doi.org/10.1007/978-3-319-62127-2_1
Download citation
DOI: https://doi.org/10.1007/978-3-319-62127-2_1
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-62126-5
Online ISBN: 978-3-319-62127-2
eBook Packages: Computer ScienceComputer Science (R0)