Covering and Packing of Rectilinear Subdivision

  • Satyabrata JanaEmail author
  • Supantha PanditEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11355)


We study a class of geometric covering and packing problems for bounded closed regions on the plane. We are given a set of axis-parallel line segments that induce a planar subdivision with bounded (rectilinear) faces. We are interested in the following problems.  
(P1) Stabbing-Subdivision:

Stab all closed bounded faces by selecting a minimum number of points in the plane.

(P2) Independent-Subdivision:

Select a maximum size collection of pairwise non-intersecting closed bounded faces.

(P3) Dominating-Subdivision:

Select a minimum size collection of bounded faces such that every other face has a non-empty intersection (i.e., sharing an edge or a vertex) with some selected face.

  We show that these problems are \(\mathsf { NP }\)-hard. We even prove that these problems are \(\mathsf { NP }\)-hard when we concentrate only on the rectangular faces of the subdivision. Further, we provide constant factor approximation algorithms for the Stabbing-Subdivision problem.


Planar subdivision Set cover Independent set Dominating set \(\mathsf { NP }\)-hard \(\mathsf {PTAS}\) 


  1. 1.
    Adamaszek, A., Wiese, A.: Approximation schemes for maximum weight independent set of rectangles. In: FOCS, pp. 400–409 (2013)Google Scholar
  2. 2.
    Chan, T.M., Har-Peled, S.: Approximation algorithms for maximum independent set of pseudo-disks. Discret. Comput. Geom. 48(2), 373–392 (2012)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Chuzhoy, J., Ene, A.: On approximating maximum independent set of rectangles. In: FOCS, pp. 820–829 (2016)Google Scholar
  4. 4.
    Czyzowicz, J., Rivera-Campo, E., Santoro, N., Urrutia, J., Zaks, J.: Guarding rectangular art galleries. Discret. Appl. Math. 50(2), 149–157 (1994)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Fowler, R.J., Paterson, M.S., Tanimoto, S.L.: Optimal packing and covering in the plane are NP-complete. Inf. Process. Lett. 12, 133–137 (1981)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Gaur, D.R., Ibaraki, T., Krishnamurti, R.: Constant ratio approximation algorithms for the rectangle stabbing problem and the rectilinear partitioning problem. J. Algorithms 43(1), 138–152 (2002)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Hochbaum, D.S., Maass, W.: Approximation schemes for covering and packing problems in image processing and VLSI. J. ACM 32(1), 130–136 (1985)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Knuth, D.E., Raghunathan, A.: The problem of compatible representatives. SIAM J. Discret. Math. 5(3), 422–427 (1992)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Korman, M., Poon, S.H., Roeloffzen, M.: Line segment covering of cells in arrangements. Inf. Process. Lett. 129, 25–30 (2018)MathSciNetCrossRefGoogle Scholar
  10. 10.
    van Leeuwen, E.J.: Optimization and approximation on systems of geometric objects. Ph.D. thesis, University of Amsterdam (2009)Google Scholar
  11. 11.
    Lichtenstein, D.: Planar formulae and their uses. SIAM J. Comput. 11(2), 329–343 (1982)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Mudgal, A., Pandit, S.: Covering, hitting, piercing and packing rectangles intersecting an inclined line. In: Lu, Z., Kim, D., Wu, W., Li, W., Du, D.-Z. (eds.) COCOA 2015. LNCS, vol. 9486, pp. 126–137. Springer, Cham (2015). Scholar
  13. 13.
    Mustafa, N.H., Raman, R., Ray, S.: Settling the APX-hardness status for geometric set cover. In: FOCS, pp. 541–550 (2014)Google Scholar
  14. 14.
    Mustafa, N.H., Ray, S.: Improved results on geometric hitting set problems. Discret. Comput. Geom. 44(4), 883–895 (2010)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Pandit, S.: Dominating set of rectangles intersecting a straight line. In: Canadian Conference on Computational Geometry, CCCG, pp. 144–149 (2017)Google Scholar
  16. 16.
    Vazirani, V.V.: Approximation Algorithms. Springer, Hecidelberg (2001)zbMATHGoogle Scholar

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Authors and Affiliations

  1. 1.Indian Statistical InstituteKolkataIndia
  2. 2.Stony Brook UniversityStony BrookUSA

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