Approximation Algorithms for Unit Disk Graphs

  • Erik Jan van Leeuwen
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3787)


We consider several graph theoretic problems on unit disk graphs (Maximum Independent Set, Minimum Vertex Cover, and Minimum (Connected) Dominating Set) relevant to mobile ad hoc networks. We propose two new notions: thickness and density. If the thickness of a unit disk graph is bounded, then the mentioned problems can be solved in polynomial time. For unit disk graphs of bounded density, we present a new asymptotic fully-polynomial approximation scheme for the considered problems. The scheme for Minimum Connected Dominating Set is the first Baker-like asymptotic FPTAS for this problem. By adapting the proof, it implies e.g. an asymptotic FPTAS for Minimum Connected Dominating Set on planar graphs.


Planar Graph Disk Center Bounded Density Unit Disk Graph Path Decomposition 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Alber, J., Fiala, J.: Geometric Separation and Exact Solutions for the Parameterized Independent Set Problem on Disk Graphs. J. Algorithms 52(2), 134–151 (2003)CrossRefMathSciNetGoogle Scholar
  2. 2.
    Alber, J., Niedermeier, R.: Improved Tree Decomposition Based Algorithms for Domination-like Problems. In: Rajsbaum, S. (ed.) LATIN 2002. LNCS, vol. 2286, pp. 613–628. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  3. 3.
    Ausiello, G., Creszenzi, P., Gambosi, G., Kann, V., Marchetti-Spaccamela, A., Protasi, M.: Complexity and Approximation - Combinatorial Optimization Problems and Their Approximability. Springer, Berlin (1999)zbMATHGoogle Scholar
  4. 4.
    Baker, B.S.: Approximation Algorithms for NP-Complete Problems on Planar Graphs. JACM 41(1), 153–180 (1994)zbMATHCrossRefGoogle Scholar
  5. 5.
    Becker, H.W.: Planar Rhyme Schemes. Bull. Am. Math. Soc. 58, 39 (1952)Google Scholar
  6. 6.
    Bodlaender, H.L.: A Tourist Guide through Treewidth. Acta Cybernetica 11(1–2), 1–22 (1993)zbMATHMathSciNetGoogle Scholar
  7. 7.
    Chan, T.M.: Polynomial-time Approximation Schemes for Packing and Piercing Fat Objects. J. Algorithms 46, 178–189 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Cheng, X., Huang, X., Li, D., Wu, W., Du, D.-Z.: A Polynomial-Time Approximation Scheme for the Minimum Connected Dominating Set in Ad Hoc Wireless Networks. Networks 42(4), 202–208 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Clark, B.N., Colbourn, C.J., Johnson, D.S.: Unit Disk Graphs. Discr. Math. 86(1–3), 165–177 (1990)zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Demaine, E.D., Hajiaghayi, M.: Bidimensionality: New Connections between FPT Algorithms and PTASs. In: SODA 2005, pp. 590–601. SIAM, Philadelphia (2005)Google Scholar
  11. 11.
    Downey, R.G., Fellows, M.R.: Parameterized Complexity. Springer, New York (1999)Google Scholar
  12. 12.
    Erlebach, T., Jansen, K., Seidel, E.: Polynomial-time Approximation Schemes for Geometric Graphs. In: SODA 2001, pp. 671–679. SIAM, Philadelphia (2001)Google Scholar
  13. 13.
    Hochbaum, D.S., Maass, W.: Approximation Schemes for Covering and Packing Problems in Image Processing and VLSI. JACM 32(1), 130–136 (1985)zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    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)zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    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)zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Matsui, T.: Approximation Algorithms for Maximum Independent Set Problems and Fractional Coloring Problems on Unit Disk Graphs. In: Akiyama, J., Kano, M., Urabe, M. (eds.) JCDCG 1998. LNCS, vol. 1763, pp. 194–200. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  17. 17.
    Nieberg, T., Hurink, J.L., Kern, W.: A Robust PTAS for Maximum Weight Independent Sets in Unit Disk Graphs. In: Hromkovič, J., Nagl, M., Westfechtel, B. (eds.) WG 2004. LNCS, vol. 3353, pp. 214–221. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  18. 18.
    Nieberg, T., Hurink, J.L.: A PTAS for the Minimum Dominating Set Problem in Unit Disk Graphs, Memorandum No. 1732, Dept. of Appl. Math., Univ. Twente, Enschede (2004)Google Scholar
  19. 19.
    Robertson, N., Seymour, P.D.: Graph Minors. I. Excluding a Forest. J. Comb. Th. B 35, 39–61 (1983)zbMATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Telle, J.A., Proskurowski, A.: Algorithms for Vertex Partitioning Problems on Partial k-Trees. SIAM J. Disc. Math. 10(4), 529–550 (1997)zbMATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    van Leeuwen, E.J.: Optimization Problems on Mobile Ad Hoc Networks – Algorithms for Disk Graphs, Master’s Thesis INF/SCR-04-32, Inst. of Information and Computing Sciences, Utrecht Univ. (2004)Google Scholar
  22. 22.
    van Leeuwen, E.J.: Approximation Algorithms for Unit Disk Graphs, Technical Report UU-CS-2004-066, Inst. of Information and Computing Sciences, Utrecht Univ. (2004)Google Scholar
  23. 23.
    Wan, P.-J., Alzoubi, K.M., Frieder, O.: Distributed Construction of Connected Dominating Set in Wireless Ad Hoc Networks. In: IEEE Infocom 2002, vol. 3, pp. 1597–1604 (2002)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Erik Jan van Leeuwen
    • 1
  1. 1.CWIAmsterdamThe Netherlands

Personalised recommendations