Domination in Geometric Intersection Graphs

  • Thomas Erlebach
  • Erik Jan van Leeuwen
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4957)


For intersection graphs of disks and other fat objects, polynomial-time approximation schemes are known for the independent set and vertex cover problems, but the existing techniques were not able to deal with the dominating set problem except in the special case of unit-size objects. We present approximation algorithms and inapproximability results that shed new light on the approximability of the dominating set problem in geometric intersection graphs. On the one hand, we show that for intersection graphs of arbitrary fat objects, the dominating set problem is as hard to approximate as for general graphs. For intersection graphs of arbitrary rectangles, we prove APX-hardness. On the other hand, we present a new general technique for deriving approximation algorithms for various geometric intersection graphs, yielding constant-factor approximation algorithms for r-regular polygons, where r is an arbitrary constant, for pairwise homothetic triangles, and for rectangles with bounded aspect ratio. For arbitrary fat objects with bounded ply, we get a (3 + ε)-approximation algorithm.


Approximation Algorithm Vertex Cover Intersection Graph Regular Polygon Unit Disk Graph 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 1.
    Alimonti, P., Kann, V.: Some APX-completeness results for cubic graphs. Theoret. Comput. Sci. 237(1-2), 123–134 (2000)CrossRefMathSciNetzbMATHGoogle Scholar
  2. 2.
    Ambühl, C., Erlebach, T., Mihalák, M., Nunkesser, M.: Constant-Factor Approximation for Minimum-Weight (Connected) Dominating Sets in Unit Disk Graphs. In: Díaz, J., Jansen, K., Rolim, J.D., Zwick, U. (eds.) Proc. APPROX-RANDOM 2006. LNCS, vol. 4110, pp. 3–14. Springer-Verlag, Berlin/Heidelberg (2006)Google Scholar
  3. 3.
    Baker, B.S.: Approximation Algorithms for NP-Complete Problems on Planar Graphs. J. ACM 41(1), 153–180 (1994)zbMATHGoogle Scholar
  4. 4.
    Bar-Yehuda, R., Even, S.: A Linear-Time Approximation Algorithm for the Weighted Vertex Cover Problem. J. Algorithms 2(2), 198–203 (1981)CrossRefMathSciNetzbMATHGoogle Scholar
  5. 5.
    Brönnimann, H., Goodrich, M.T.: Almost Optimal Set Covers in Finite VC-Dimension. Discrete Comput. Geometry 14(4), 463–479 (1995)CrossRefzbMATHGoogle Scholar
  6. 6.
    Chan, T.M.: Polynomial-time Approximation Schemes for Packing and Piercing Fat Objects. J. Algorithms 46(2), 178–189 (2003)CrossRefMathSciNetzbMATHGoogle Scholar
  7. 7.
    Chang, M.-S.: Efficient Algorithms for the Domination Problems on Interval and Circular-Arc Graphs. SIAM J. Comput. 27(6), 1671–1694 (1998)CrossRefMathSciNetzbMATHGoogle Scholar
  8. 8.
    Chlebík, M., Chlebíková, J.: Approximation Hardness of Dominating Set Problems. In: Albers, S., Radzik, T. (eds.) ESA 2004. LNCS, vol. 3221, pp. 192–203. Springer-Verlag, Berlin/Heidelberg (2004)Google Scholar
  9. 9.
    Chlebík, M., Chlebíková, J.: The Complexity of Combinatorial Optimization Problems on d-Dimensional Boxes. SIAM J. Discrete Math. 21(1), 158–169 (2007)CrossRefMathSciNetzbMATHGoogle Scholar
  10. 10.
    Clark, B.N., Colbourn, C.J., Johnson, D.S.: Unit Disk Graphs. Discrete Math. 86(1–3), 165–177 (1990)CrossRefMathSciNetzbMATHGoogle Scholar
  11. 11.
    Clarkson, K.L., Varadarajan, K.R.: Improved Approximation Algorithms for Geometric Set Cover. Discrete Comput. Geometry 37(1), 43–58 (2007)CrossRefMathSciNetzbMATHGoogle Scholar
  12. 12.
    Efrat, A., Sharir, M.: The Complexity of the Union of Fat Objects in the Plane. Discrete Comput. Geometry 23(2), 171–189 (2000)CrossRefMathSciNetzbMATHGoogle Scholar
  13. 13.
    Erlebach, T., Jansen, K., Seidel, E.: Polynomial-time Approximation Schemes for Geometric Intersection Graphs. SIAM J. Comput. 34(6), 1302–1323 (2005)CrossRefMathSciNetzbMATHGoogle Scholar
  14. 14.
    Even, G., Rawitz, D., Sharar, S.: Hitting Sets when the VC-Dimension is Small. Inform. Process. Lett. 95(2), 358–362 (2005)CrossRefMathSciNetGoogle Scholar
  15. 15.
    Feige, U.: A Threshold of ln n for Approximating Set Cover. J. ACM 45(4), 634–652 (1998)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Hochbaum, D.S.: Approximation Algorithms for the Set Covering and Vertex Cover Problems. SIAM J. Comput. 11(3), 555–556 (1982)CrossRefMathSciNetzbMATHGoogle Scholar
  17. 17.
    Hochbaum, D.S., Maass, W.: Approximation Schemes for Covering and Packing Problems in Image Processing and VLSI. J. ACM 32(1), 130–136 (1985)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Hunt III, D.B., Marathe, M.V., Radhakrishnan, V., Ravi, S.S., Rosenkrantz, D.J., Stearns, R.E.: NC-Approximation Schemes for NP- and PSPACE-Hard Problems for Geometric Graphs. J. Algorithms 26(2), 238–274 (1998)CrossRefMathSciNetzbMATHGoogle Scholar
  19. 19.
    Kedem, K., Livne, R., Pach, J., Sharir, M.: On the Union of Jordan Regions and Collision-Free Translational Motion Amidst Polygonal Obstacles. Discrete Comput. Geometry 1, 59–70 (1986)CrossRefMathSciNetzbMATHGoogle Scholar
  20. 20.
    Kim, S.-J., Kostochka, A., Nakprasit, K.: On the Chromatic Number of Intersection Graphs of Convex Sets in the Plane. Electr. J. Combinatorics 11, #R52 (2004)Google Scholar
  21. 21.
    Koebe, P.: Kontaktprobleme der konformen Abbildung. Ber. Ver. Sächs. Ak. Wiss. Leipzig, Math.-Phys. Kl. 88, 141–164 (1936)Google Scholar
  22. 22.
    Marathe, M.V., Breu, H., Hunt III, H.B., Ravi, S.S., Rosenkrantz, D.J.: Simple Heuristics for Unit Disk Graphs. Networks 25, 59–68 (1995)CrossRefMathSciNetzbMATHGoogle Scholar
  23. 23.
    Marx, D.: Parameterized Complexity of Independence and Domination on Geometric Graphs. In: Bodlaender, H.L., Langston, M.A. (eds.) IWPEC 2006. LNCS, vol. 4169, pp. 154–165. Springer-Verlag, Berlin/Heidelberg (2006)CrossRefGoogle Scholar
  24. 24.
    Miller, G.L., Teng, S.-H., Thurston, W., Vavasis, S.A.: Separators for Sphere-Packings and Nearest Neighbor Graphs. J. ACM 44(1), 1–29 (1997)MathSciNetzbMATHGoogle Scholar
  25. 25.
    Nieberg, T., Hurink, J.L.: A PTAS for the Minimum Dominating Set Problem in Unit Disk Graphs. In: Erlebach, T., Persinao, G. (eds.) WAOA 2005. LNCS, vol. 3879, pp. 296–306. Springer-Verlag, Berlin/Heidelberg (2006)CrossRefGoogle Scholar
  26. 26.
    van Leeuwen, E.J.: Better Approximation Schemes for Disk Graphs. In: Arge, L., Freivalds, R. (eds.) SWAT 2006. LNCS, vol. 4059, pp. 316–327. Springer-Verlag, Berlin/Heidelberg (2006)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Thomas Erlebach
    • 1
  • Erik Jan van Leeuwen
    • 2
  1. 1.Department of Computer ScienceUniversity of LeicesterLeicesterUK
  2. 2.CWISJ AmsterdamThe Netherlands

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