Advertisement

Independent and Hitting Sets of Rectangles Intersecting a Diagonal Line

  • José R. Correa
  • Laurent Feuilloley
  • José A. Soto
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8392)

Abstract

Finding a maximum independent set of a given family of axis-parallel rectangles is a basic problem in computational geometry and combinatorics. This problem has attracted significant attention since the sixties, when Wegner conjectured that the corresponding duality gap, i.e., the maximum possible ratio between the maximum independent set and the minimum hitting set, is bounded by a universal constant. In this paper we improve upon recent results of Chepoi and Felsner and prove that when the given family of rectangles is intersected by a diagonal, this ratio is between 2 and 4. For the upper bound we derive a simple combinatorial argument that first allows us to reprove results of Hixon, and Chepoi and Felsner and then we adapt this idea to obtain the improved bound in the diagonal intersecting case. From a computational complexity perspective, although for general rectangle families the problem is known to be NP-hard, we derive an O(n 2)-time algorithm for the maximum weight independent set when, in addition to intersecting a diagonal, the rectangles intersect below it. This improves and extends a classic result of Lubiw. As a consequence, we obtain a 2-approximation algorithm for the maximum weight independent set of rectangles intersecting a diagonal.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Adamaszek, A., Wiese, A.: Approximation Schemes for Maximum Weight Independent Set of Rectangles. In: FOCS 2013 (2013)Google Scholar
  2. 2.
    Aronov, B., Ezra, E., Sharir, M.: Small-size ε-nets for axis-parallel rectangles and boxes. SIAM J. Comp. 39, 3248–3282 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Chalermsook, P., Chuzhoy, J.: Maximum independent set of rectangles. In: SODA 2009 (2009)Google Scholar
  4. 4.
    Chan, T.M., Har-Peled, S.: Approximation algorithms for maximum independent set of pseudo-disks. In: SoCG 2009 (2009)Google Scholar
  5. 5.
    Chepoi, V., Felsner, S.: Approximating hitting sets of axis-parallel rectangles intersecting a monotone curve. Computational Geometry 46, 1036–1041 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Cibulka, J., Hladký, J., Kazda, A., Lidický, B., Ondráčková, E., Tancer, M., Jelínek, V.: Personal Communication (2011)Google Scholar
  7. 7.
    Correa, J.R., Feuilloley, L., Pérez-Lantero, P., Soto, J.A.: Independent and Hitting Sets of Rectangles Intersecting a Diagonal Line: Algorithms and Complexity. arXiv:1309.6659Google Scholar
  8. 8.
    Fon-Der-Flaass, D.G., Kostochka, A.V.: Covering boxes by points. Disc. Math. 120, 269–275 (1993)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Gyárfás, A., Lehel, J.: Covering and coloring problems for relatives of intervals. Disc. Math. 55, 167–180 (1985)CrossRefzbMATHGoogle Scholar
  10. 10.
    Hixon, T.S.: Hook graphs and more: Some contributions to geometric graph theory. Master’s thesis, Technische Universitat Berlin (2013)Google Scholar
  11. 11.
    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)CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Hsiao, J.Y., Tang, C.Y., Chang, R.S.: An efficient algorithm for finding a maximum weight 2-independent set on interval graphs. Inf. Process. Lett. 43, 229–235 (1992)CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Imai, H., Asano, T.: Finding the connected components and a maximum clique of an intersection graph of rectangles in the plane. J. of Algorithms 4, 310–323 (1983)CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Károlyi, G., Tardos, G.: On point covers of multiple intervals and axis-parallel rectangles. Combinatorica 16, 213–222 (1996)CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    Lubiw, A.: A weighted min-max relation for intervals. J. Comb. Theory, Ser. B 53, 151–172 (1991)CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    Mustafa, N.H., Ray, S.: Improved results on geometric hitting set problems. Discrete Comput. Geom. 44, 883–895 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    Soto, J.A., Telha, C.: Jump Number of Two-Directional Orthogonal Ray Graphs. In: Günlük, O., Woeginger, G.J. (eds.) IPCO 2011. LNCS, vol. 6655, pp. 389–403. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  18. 18.
    Soto, M., Thraves, C.: (c-)And graphs - more than intersection, more than geometric. arXiv:1306.1957 (2013) (submitted)Google Scholar
  19. 19.
    Wegner, G.: Über eine kombinatorisch-geometrische frage von hadwiger und debrunner. Israel J. of Mathematics 3, 187–198 (1965)CrossRefzbMATHMathSciNetGoogle Scholar
  20. 20.
    Yannakakis, M.: The complexity of the partial order dimension problem. SIAM J. Alg. Discr. Meth. 3, 351–358 (1982)CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • José R. Correa
    • 1
  • Laurent Feuilloley
    • 2
  • José A. Soto
    • 3
  1. 1.Department of Industrial EngineeringUniversidad de ChileChile
  2. 2.Department of Computer ScienceENS CachanFrance
  3. 3.Department of Mathematical Engineering and Center for Mathematical ModelingUniversidad de ChileChile

Personalised recommendations