Skip to main content
SpringerLink
Log in

Minimum vertex cover in ball graphs through local search

  • Published:
Journal of Global Optimization Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1

Similar content being viewed by others

References

  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)

  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. 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. Cormen, T.H., Leiserson, C.E., Rivest, R.L., Stein, C.: Introduction to algorithms, 2nd edn. The MIT Press, Cambridge (2001)

    Google Scholar 

  5. Dijdjev, H.N., Gilbert, J.R.: Separators in graphs with negative and multiple vertex weights. Algorithmica 23, 57–71 (1999)

    Article  Google Scholar 

  6. Dinur, I., Safra, S.: On the hardness of approximating minimum vertex cover. Ann. Math. 162, 439–485 (2005)

    Article  Google Scholar 

  7. Erlebach, T., Jansen, K., Seidel, E.: Polynomial-time approximation schemes for geometric intersection graphs. SIAM J. Comput. 34, 1302–1323 (2005)

    Article  Google Scholar 

  8. Erlebach, T., van Leeuwen, E.J.: Domination in geometric intersection graphs, LATIN 2008. Lecture Notes in Computer ScienceVolume 4957, 747–758 (2008)

  9. Frederickson, G.N.: Fast algorithms for shortest paths in planar graphs, with applications. SIAM J. Comput. 16, 1004–1022 (1987)

    Article  Google Scholar 

  10. Garey, M.R., Johnson, D.S.: Computers and intractability: a guide to the theory of NP-completeness. W.H. Freeman, San Francisco (1979)

  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)

  12. Hochbaum, D.S., Maass, W.: Approximation schemes for covering and packing problems in image processing and VLSI. J. ACM 32, 130–136 (1985)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  14. Karakostas, G.: A better approximation ratio for the vertex cover problem, ICALP’05. Lecture Notes in Computer Science 3580, 1043–1050 (2005)

  15. Koebe, P.: Kontaktprobleme der konformen Abbildung. Ber. Verh. Sächs. Akademie der Wissenschaften Leipzig Math. Phys. Klasse 88, 141–164 (1936)

    Google Scholar 

  16. Lipton, R.J., Tarjan, R.E.: A separator theorem for planar graphs. SIAM J. Appl. Math. 36, 177–189 (1979)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  19. Monien, B., Speckenmeyer, E.: Ramsey numbers and an approximation algorithm for the vertex cover problem. Acta Informatica 22, 115–123 (1985)

    Article  Google Scholar 

  20. Mustafa, N.H., Ray, S.: PTAS for geometric hitting set problems via local search, SCG’09, 17–22 (2009)

  21. Pach, J., Agarwal, P.K.: Combinatorial geometry. Wiley-Interscience Publication, New York (1995)

    Book  Google Scholar 

  22. Papadimitriou, C.H.: Combinatorial optimization. Prentice-Hall, Inc, Englewood Cliffs, NJ (1982)

    Google Scholar 

  23. Papadimitriou, C.H., Yannakakis, M.: Optimization, approximation, and complexity classes. J. Comput. Syst. Sci. 43, 425–440 (1991)

    Article  Google Scholar 

  24. Vazirani, V.V.: Approximation algorithms. Springer, Berlin (2001)

    Google Scholar 

Download references

Acknowledgments

The work is supported by NSFC (61222201) and SRFDP (20126501110001) Xinjiang Talent Project (2013711011), and by National Science Foundation of USA under grants CNS0831579 and CCF0728851.

Author information

Authors and Affiliations

  1. College of Mathematics and System Sciences, Xinjiang University Urumqi, Xinjiang, 830046, People’s Republic of China

    Zhao Zhang

  2. Department of Computer Science, University of Texas at Dallas, Richardson, TX, 75080, USA

    Weili Wu, Lidan Fan & Ding-Zhu Du

Authors
  1. Zhao Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Weili Wu
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Lidan Fan
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Ding-Zhu Du
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Zhao Zhang.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Zhang, Z., Wu, W., Fan, L. et al. Minimum vertex cover in ball graphs through local search. J Glob Optim 59, 663–671 (2014). https://doi.org/10.1007/s10898-013-0116-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10898-013-0116-4

Keywords

Search

Navigation