Skip to main content
SpringerLink
Log in

Algorithm and Hardness Results on Liar’s Dominating Set and \(\varvec{k}\)-tuple Dominating Set

  • Conference paper
  • First Online:
Combinatorial Algorithms (IWOCA 2019)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 11638))

Included in the following conference series:

Abstract

Given a graph \(G=(V,E)\), the dominating set problem asks for a minimum subset of vertices \(D\subseteq V\) such that every vertex \(u\in V\setminus D\) is adjacent to at least one vertex \(v\in D\). That is, the set D satisfies the condition that \(|N[v]\cap D|\ge 1\) for each \(v\in V\), where N[v] is the closed neighborhood of v. In this paper, we study two variants of the classical dominating set problem: \(\varvec{k}\)-tuple dominating set (k-DS) problem and Liar’s dominating set (LDS) problem, and obtain several algorithmic and hardness results. On the algorithmic side, we present a constant factor (\(\frac{11}{2}\))-approximation algorithm for the Liar’s dominating set problem on unit disk graphs. Then, we design a polynomial time approximation scheme (PTAS) for the \(\varvec{k}\)-tuple dominating set problem on unit disk graphs. On the hardness side, we show a \(\varOmega (n^2)\) bits lower bound for the space complexity of any (randomized) streaming algorithm for Liar’s dominating set problem as well as for the \(\varvec{k}\)-tuple dominating set problem. Furthermore, we prove that the Liar’s dominating set problem on bipartite graphs is W[2]-hard.

S. Bhore—Supported by the Austrian Science Fund (FWF) under project number P31119.

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

Access this chapter

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 59.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 79.99
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

Institutional subscriptions

Notes

  1. 1.

    Proof of results labeled with \([\star ]\) have been deferred to the full version [1] due to space constraint.

  2. 2.

    By randomized algorithm we mean that the algorithm should succeed with probability \(\ge \frac{2}{3}\).

References

  1. Banerjee, S., Bhore, S.: Algorithm and hardness results on liar’s dominating set and \(k\)-tuple dominating set. CoRR, abs/1902.11149 (2019)

    Google Scholar 

  2. Bishnu, A., Ghosh, A., Paul, S.: Linear kernels for k-tuple and liar’s domination in bounded genus graphs. Discrete Appl. Math. 231, 67–77 (2017)

    Article  MathSciNet  Google Scholar 

  3. Chang, G.J.: Algorithmic aspects of domination in graphs. In: Du, D.Z., Pardalos, P.M. (eds.) Handbook of Combinatorial Optimization, pp. 1811–1877. Springer, Boston (1998). https://doi.org/10.1007/978-1-4613-0303-9_28

    Chapter  Google Scholar 

  4. Chellali, M., Favaron, O., Hansberg, A., Volkmann, L.: k-domination and k-independence in graphs: a survey. Graphs Comb. 28(1), 1–55 (2012)

    Article  MathSciNet  Google Scholar 

  5. Fink, J.F., Jacobson, M.S.: n-domination in graphs. In: Graph theory with Applications to Algorithms and Computer Science, pp. 283–300. Wiley, Hoboken (1985)

    Google Scholar 

  6. Garey, M.R., Johnson, D.S.: Computers and Intractability, vol. 29. W. H. Freeman, New York (2002)

    Google Scholar 

  7. Harary, F., Haynes, T.W.: Double domination in graphs. Ars Comb. 55, 201–214 (2000)

    MathSciNet  MATH  Google Scholar 

  8. Haynes, T.W., Hedetniemi, S., Slater, P.: Fundamentals of Domination in Graphs. CRC Press, Boca Raton (1998)

    MATH  Google Scholar 

  9. Jallu, R.K., Jena, S.K., Das, G.K.: Liar’s dominating set in unit disk graphs. In: Wang, L., Zhu, D. (eds.) COCOON 2018. LNCS, vol. 10976, pp. 516–528. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-94776-1_43

    Chapter  Google Scholar 

  10. Klasing, R., Laforest, C.: Hardness results and approximation algorithms of k-tuple domination in graphs. Inf. Process. Lett. 89(2), 75–83 (2004)

    Article  MathSciNet  Google Scholar 

  11. Kushilevitz, E., Nisan, N.: Communication Complexity. Cambridge University Press, Cambridge (1997)

    MATH  Google Scholar 

  12. Liao, C.S., Chang, G.J.: k-tuple domination in graphs. Inf. Process. Lett. 87(1), 45–50 (2003)

    Article  MathSciNet  Google Scholar 

  13. Nieberg, T., Hurink, J.: A PTAS for the minimum dominating set problem in unit disk graphs. In: Erlebach, T., Persinao, G. (eds.) WAOA 2005. LNCS, vol. 3879, pp. 296–306. Springer, Heidelberg (2006). https://doi.org/10.1007/11671411_23

    Chapter  Google Scholar 

  14. Panda, B.S., Paul, S.: Liar’s domination in graphs: complexity and algorithm. Discrete Appl. Math. 161(7–8), 1085–1092 (2013)

    Article  MathSciNet  Google Scholar 

  15. Rautenbach, D., Volkmann, L.: New bounds on the k-domination number and the k-tuple domination number. Appl. Math. Lett. 20(1), 98–102 (2007)

    Article  MathSciNet  Google Scholar 

  16. Roden, M.L., Slater, P.J.: Liar’s domination in graphs. Discrete Math. 309(19), 5884–5890 (2009)

    Article  MathSciNet  Google Scholar 

  17. Slater, P.J.: Liar’s domination. Networks 54(2), 70–74 (2009)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The authors would like to thank Rajesh Chitnis and M. S. Ramanujan for interesting discussions during various stages of the research.

Author information

Authors and Affiliations

  1. Department of Computer Science, Hebrew University of Jerusalem, Jerusalem, Israel

    Sandip Banerjee

  2. Algorithms and Complexity Group, TU Wien, Vienna, Austria

    Sujoy Bhore

Authors
  1. Sandip Banerjee
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Sujoy Bhore
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Sujoy Bhore .

Editor information

Editors and Affiliations

  1. Arizona State University, Tempe, AZ, USA

    Charles J. Colbourn

  2. University of Pisa, Pisa, Italy

    Roberto Grossi

  3. University of Pisa, Pisa, Italy

    Nadia Pisanti

Rights and permissions

Reprints and permissions

Copyright information

© 2019 Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Banerjee, S., Bhore, S. (2019). Algorithm and Hardness Results on Liar’s Dominating Set and \(\varvec{k}\)-tuple Dominating Set. In: Colbourn, C., Grossi, R., Pisanti, N. (eds) Combinatorial Algorithms. IWOCA 2019. Lecture Notes in Computer Science(), vol 11638. Springer, Cham. https://doi.org/10.1007/978-3-030-25005-8_5

Download citation

  • DOI: https://doi.org/10.1007/978-3-030-25005-8_5

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-25004-1

  • Online ISBN: 978-3-030-25005-8

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation