Journal of Global Optimization

, Volume 59, Issue 2–3, pp 663–671 | Cite as

Minimum vertex cover in ball graphs through local search

Article

Abstract

Using local search method, this paper provides a polynomial time approximation scheme for the minimum vertex cover problem on \(d\)-dimensional ball graphs where \(d \ge 3\). The key to the proof is a new separator theorem for ball graphs in higher dimensional space.

Keywords

Vertex cover Ball graph Local search Separator theorem 

References

  1. 1.
    Baker, B.S.: Approximation algorithms for NP-complete problems on planar graphs, In proceedings of the 24th annual IEEE symposium on foundations of computer science, Tucson, Ariz, Nov. 7–9, IEEE, New York: 254–273. J. ACM 41(1994), 153–180 (1983)Google Scholar
  2. 2.
    Bar-Yehuda, R., Even, S.: A local-ration theorem for approximating the weighted vertex cover problem, analysis and design of algorithms for combinatorial problems. Ann. Discret. Math. 25, 27–46 (1985)Google Scholar
  3. 3.
    Chan, T.M., Har-Peled, S.: Approximation algorithms for maximum independent set of pseudo-disks. Discret. Comput. Geom. 48(2), 373–392 (2009)Google Scholar
  4. 4.
    Cormen, T.H., Leiserson, C.E., Rivest, R.L., Stein, C.: Introduction to algorithms, 2nd edn. The MIT Press, Cambridge (2001)Google Scholar
  5. 5.
    Dijdjev, H.N., Gilbert, J.R.: Separators in graphs with negative and multiple vertex weights. Algorithmica 23, 57–71 (1999)CrossRefGoogle Scholar
  6. 6.
    Dinur, I., Safra, S.: On the hardness of approximating minimum vertex cover. Ann. Math. 162, 439–485 (2005)CrossRefGoogle Scholar
  7. 7.
    Erlebach, T., Jansen, K., Seidel, E.: Polynomial-time approximation schemes for geometric intersection graphs. SIAM J. Comput. 34, 1302–1323 (2005)CrossRefGoogle Scholar
  8. 8.
    Erlebach, T., van Leeuwen, E.J.: Domination in geometric intersection graphs, LATIN 2008. Lecture Notes in Computer ScienceVolume 4957, 747–758 (2008)Google Scholar
  9. 9.
    Frederickson, G.N.: Fast algorithms for shortest paths in planar graphs, with applications. SIAM J. Comput. 16, 1004–1022 (1987)CrossRefGoogle Scholar
  10. 10.
    Garey, M.R., Johnson, D.S.: Computers and intractability: a guide to the theory of NP-completeness. W.H. Freeman, San Francisco (1979)Google Scholar
  11. 11.
    Gibson, M., Pirwani, I.A.: Algorithms for dominating set in disk graphs: breaking the \(\log n\) barrier, ESA 2010. Lecture Notes in Computer Science 6346, 243–254 (2010)Google Scholar
  12. 12.
    Hochbaum, D.S., Maass, W.: Approximation schemes for covering and packing problems in image processing and VLSI. J. ACM 32, 130–136 (1985)CrossRefGoogle Scholar
  13. 13.
    Hunt, H.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, 238–274 (1998)CrossRefGoogle Scholar
  14. 14.
    Karakostas, G.: A better approximation ratio for the vertex cover problem, ICALP’05. Lecture Notes in Computer Science 3580, 1043–1050 (2005)Google Scholar
  15. 15.
    Koebe, P.: Kontaktprobleme der konformen Abbildung. Ber. Verh. Sächs. Akademie der Wissenschaften Leipzig Math. Phys. Klasse 88, 141–164 (1936)Google Scholar
  16. 16.
    Lipton, R.J., Tarjan, R.E.: A separator theorem for planar graphs. SIAM J. Appl. Math. 36, 177–189 (1979)CrossRefGoogle Scholar
  17. 17.
    Marathe, M.V., Breu, H., Hunt, H.B., Ravi, S.S., Rosenkrantz, D.J.: Simple heuristics for unit disk graphs. Networks 25, 59–68 (1995)CrossRefGoogle Scholar
  18. 18.
    Miller, G.L., Teng, S., Thurston, W., Vavasis, S.A.: Separators for sphere-packings and nearest neigbhor graphs. J. ACM 44, 1–29 (1997)CrossRefGoogle Scholar
  19. 19.
    Monien, B., Speckenmeyer, E.: Ramsey numbers and an approximation algorithm for the vertex cover problem. Acta Informatica 22, 115–123 (1985)CrossRefGoogle Scholar
  20. 20.
    Mustafa, N.H., Ray, S.: PTAS for geometric hitting set problems via local search, SCG’09, 17–22 (2009)Google Scholar
  21. 21.
    Pach, J., Agarwal, P.K.: Combinatorial geometry. Wiley-Interscience Publication, New York (1995)CrossRefGoogle Scholar
  22. 22.
    Papadimitriou, C.H.: Combinatorial optimization. Prentice-Hall, Inc, Englewood Cliffs, NJ (1982)Google Scholar
  23. 23.
    Papadimitriou, C.H., Yannakakis, M.: Optimization, approximation, and complexity classes. J. Comput. Syst. Sci. 43, 425–440 (1991)CrossRefGoogle Scholar
  24. 24.
    Vazirani, V.V.: Approximation algorithms. Springer, Berlin (2001)Google Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.College of Mathematics and System SciencesXinjiang University UrumqiXinjiangPeople’s Republic of China
  2. 2.Department of Computer ScienceUniversity of Texas at DallasRichardsonUSA

Personalised recommendations