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Better Approximation Schemes for Disk Graphs

  • Erik Jan van Leeuwen
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4059)

Abstract

We consider Maximum Independent Set and Minimum Vertex Cover on disk graphs. We propose an asymptotic FPTAS for Minimum Vertex Cover on disk graphs of bounded ply. This scheme can be extended to an EPTAS on arbitrary disk graphs, improving on the previously known PTAS [8]. We introduce the notion of level density for disk graphs, which is a generalization of the notion of ply. We give an asymptotic FPTAS for Maximum Independent Set on disk graphs of bounded level density, which is also a PTAS on arbitrary disk graphs. The schemes are a geometric generalization of Baker’s EPTASs for planar graphs [3].

Keywords

Planar Graph Approximation Scheme Vertex Cover Intersection Graph 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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References

  1. 1.
    Agarwal, P.K., Overmars, M., Sharir, M.: Computing maximally separated sets in the plane and independent sets in the intersection graph of unit disks. In: SODA 2004, pp. 516–525. SIAM, Philadelphia (2004)Google Scholar
  2. 2.
    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
  3. 3.
    Baker, B.S.: Approximation Algorithms for NP-Complete Problems on Planar Graphs. J. ACM 41(1), 153–180 (1994)zbMATHCrossRefGoogle Scholar
  4. 4.
    Chan, T.M.: Polynomial-time Approximation Schemes for Packing and Piercing Fat Objects. J. Algorithms 46(2), 178–189 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Chan, T.M.: A Note on Maximum Independent Sets in Rectangle Intersection Graphs. Inf. Proc. Let. 89(1), 19–23 (2004)zbMATHCrossRefGoogle Scholar
  6. 6.
    Clark, B.N., Colbourn, C.J., Johnson, D.S.: Unit Disk Graphs. Discr. Math. 86(1–3), 165–177 (1990)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Eppstein, D., Miller, G.L., Teng, S.-H.: A Deterministic Linear Time Algorithm for Geometric Separators and its Applications. Fund. Inform. 22(4), 309–329 (1995)zbMATHMathSciNetGoogle Scholar
  8. 8.
    Erlebach, T., Jansen, K., Seidel, E.: Polynomial-time Approximation Schemes for Geometric Intersection Graphs. SIAM J. Computing 34(6), 1302–1323 (2005)zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Hochbaum, D.S., Maass, W.: Approximation Schemes for Covering and Packing Problems in Image Processing and VLSI. J. ACM 32(1), 130–136 (1985)zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    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
  11. 11.
    Koebe, P.: Kontaktprobleme der konformen Abbildung. Ber. Ver. Sächs. Ak. Wiss. Leipzig, Math.-Phys. Kl. 88, 141–164 (1936)Google Scholar
  12. 12.
    Malesińska, E.: Graph-Theoretical Models for Frequency Assignment Problems, PhD Thesis, Technical University of Berlin, Berlin (1997)Google Scholar
  13. 13.
    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
  14. 14.
    Marx, D.: Efficient Approximation Schemes for Geometric Problems? In: Brodal, G.S., Leonardi, S. (eds.) ESA 2005. LNCS, vol. 3669, pp. 448–459. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  15. 15.
    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
  16. 16.
    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)zbMATHCrossRefMathSciNetGoogle 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.
    Robertson, N., Seymour, P.D.: Graph Minors. I. Excluding a Forest. J. Comb. Th. B 35, 39–61 (1983)zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    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
  20. 20.
    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
  21. 21.
    van Leeuwen, E.J.: Approximation Algorithms for Unit Disk Graphs. In: Kratsch, D. (ed.) WG 2005. LNCS, vol. 3787, pp. 351–361. Springer, Heidelberg (2005)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

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

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