Skip to main content
SpringerLink
Log in
Book cover

International Conference on Combinatorial Optimization and Applications

COCOA 2017: Combinatorial Optimization and Applications pp 348–361Cite as

Parameterized Approximation Algorithms for Some Location Problems in Graphs

  • Conference paper
  • First Online:

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10628))

Abstract

We develop efficient parameterized, with additive error, approximation algorithms for the (Connected) r-Domination problem and the (Connected) p-Center problem for unweighted and undirected graphs. Given a graph G, we show how to construct a (connected) \(\big (r + \mathcal {O}(\mu ) \big )\)-dominating set D with \(|D| \le |D^*|\) efficiently. Here, \(D^*\) is a minimum (connected) r-dominating set of G and \(\mu \) is our graph parameter, which is the tree-breadth or the cluster diameter in a layering partition of G. Additionally, we show that a \(+ \mathcal {O}(\mu )\)-approximation for the (Connected) p-Center problem on G can be computed in polynomial time. Our interest in these parameters stems from the fact that in many real-world networks, including Internet application networks, web networks, collaboration networks, social networks, biological networks, and others, and in many structured classes of graphs these parameters are small constants.

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

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

References

  1. Abboud, A., Williams, V.V., Wang, J.: Approximation and fixed parameter subquadratic algorithms for radius and diameter in sparse graphs. In: Proceedings of the 27th Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 377–391 (2016)

    Google Scholar 

  2. Abu-Ata, M., Dragan, F.F.: Metric tree-like structures in real-life networks: an empirical study. Networks 67(1), 49–68 (2016)

    Article  MathSciNet  Google Scholar 

  3. Brandstädt, A., Chepoi, V., Dragan, F.F.: The algorithmic use of hypertree structure and maximum neighbourhood orderings. Discrete Appl. Math. 82(1–3), 43–77 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  4. Brandstädt, A., Chepoi, V., Dragan, F.F.: Distance approximating trees for chordal and dually chordal graphs. J. Algorithms 30, 166–184 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  5. Chalermsook, P., Cygan, M., Kortsarz, G., Laekhanukit, B., Manurangsi, P., Nanongkai, D., Trevisan, D.: From Gap-ETH to FPT-Inapproximability: Clique, Dominating Set, and More Manuscript, CoRR abs/1708.04218 (2017)

    Google Scholar 

  6. Chen, Y., Lin, B.: The Constant Inapproximability of the Parameterized Dominating Set Problem, Manuscript, CoRR abs/1511.00075 (2015)

    Google Scholar 

  7. Chepoi, V., Dragan, F.F.: A note on distance approximating trees in graphs. Eur. J. Comb. 21, 761–766 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  8. Chepoi, V.D., Dragan, F.F., Estellon, B., Habib, M., Vaxes, Y.: Diameters, centers, and approximating trees of \(\delta \)-hyperbolic geodesic spaces and graphs. In: Proceedings of the 24th Annual ACM Symposium on Computational Geometry, SoCG 2008, pp. 59–68 (2008)

    Google Scholar 

  9. Chepoi, V., Estellon, B.: Packing and covering \(\delta \)-hyperbolic spaces by balls. In: Charikar, M., Jansen, K., Reingold, O., Rolim, J.D.P. (eds.) APPROX/RANDOM -2007. LNCS, vol. 4627, pp. 59–73. Springer, Heidelberg (2007). https://doi.org/10.1007/978-3-540-74208-1_5

    Chapter  Google Scholar 

  10. Chlebík, M., Chlebíková, J.: Approximation hardness of dominating set problems in bounded degree graphs. Inf. Comput. 206, 1264–1275 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  11. Cormen, T.H., Leiserson, C.E., Rivest, R.L., Stein, C.: Introduction to Algorithms, 3rd edn. MIT Press, Cambridge (2009)

    MATH  Google Scholar 

  12. Dourisboure, Y., Dragan, F.F., Gavoille, C., Yan, C.: Spanners for bounded tree-length graphs. Theor. Comput. Sci. 383(1), 34–44 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  13. Downey, R.G., Fellows, M.R.: Parameterized Complexity. Springer, New York (1999)

    Book  MATH  Google Scholar 

  14. Dragan, F.F.: HT-graphs: centers, connected \(r\)-domination and Steiner trees. Comput. Sci. J. Moldova 1(2), 64–83 (1993)

    MathSciNet  Google Scholar 

  15. Edwards, K., Kennedy, S., Saniee, I.: Fast approximation algorithms for p-centers in large \(\delta \)-hyperbolic graphs. In: Bonato, A., Graham, F.C., Prałat, P. (eds.) WAW 2016. LNCS, vol. 10088, pp. 60–73. Springer, Cham (2016). https://doi.org/10.1007/978-3-319-49787-7_6

    Chapter  Google Scholar 

  16. Escoffier, B., Paschos, V.T.: Completeness in approximation classes beyond APX. Theor. Comput. Sci. 359(1–3), 369–377 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  17. Gonzalez, T.: Clustering to minimize the maximum intercluster distance. Theor. Comput. Sci. 38, 293–306 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  18. Guha, S., Khuller, S.: Approximation algorithms for connected dominating sets. Algorithmica 20(4), 374–387 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  19. Leitert, A., Dragan, F.F.: Parametrized Approximation Algorithms for some Location Problems in Graphs, Manuscript, CoRR abs/1706.07475 (2017)

    Google Scholar 

  20. Niedermeier, R.: Invitation to Fixed-Parameter Algorithms. Oxford Lecture Series in Mathematics and Its Applications. Oxford University Press, Oxford (2006)

    Book  MATH  Google Scholar 

  21. Raz, R., Safra, S.: A sub-constant error-probability low-degree test, and sub-constant error-probability PCP characterization of NP. In: Proceedings of the 29th Annual ACM Symposium on Theory of Computing, pp. 475–484 (1997)

    Google Scholar 

  22. Yen, W.C.-K., Chen, C.-T.: The \(p\)-center problem with connectivity constraint. Appl. Math. Sci. 1(27), 1311–1324 (2007)

    MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Computer Science, Central Washington University, Ellensburg, WA, USA

    Arne Leitert

  2. Department of Computer Science, Kent State University, Kent, OH, USA

    Feodor F. Dragan

Authors
  1. Arne Leitert
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Feodor F. Dragan
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Arne Leitert .

Editor information

Editors and Affiliations

  1. Shanghai Jiao Tong University, Shanghai, China

    Xiaofeng Gao

  2. Harbin Institute of Technology, Shenzhen, China

    Hongwei Du

  3. Kennesaw State University, Kennesaw, Georgia, USA

    Meng Han

Rights and permissions

Reprints and permissions

Copyright information

© 2017 Springer International Publishing AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Leitert, A., Dragan, F.F. (2017). Parameterized Approximation Algorithms for Some Location Problems in Graphs. In: Gao, X., Du, H., Han, M. (eds) Combinatorial Optimization and Applications. COCOA 2017. Lecture Notes in Computer Science(), vol 10628. Springer, Cham. https://doi.org/10.1007/978-3-319-71147-8_24

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-71147-8_24

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-71146-1

  • Online ISBN: 978-3-319-71147-8

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation