Skip to main content
SpringerLink
Log in

Approximability of the Minimum Weighted Doubly Resolving Set Problem

  • Conference paper
Computing and Combinatorics (COCOON 2014)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 8591))

Included in the following conference series:

Abstract

Locating source of diffusion in networks is crucial for controlling and preventing epidemic risks. It has been studied under various probabilistic models. In this paper, we study source location from a deterministic point of view by modeling it as the minimum weighted doubly resolving set problem, which is a strengthening of the well-known metric dimension problem.

Let G be a vertex weighted undirected graph on n vertices. A vertex subset S of G is a doubly resolving set (DRS) of G if for every pair of vertices u,v in G, there exist x,y ∈ S such that the difference of distances (in terms of number of edges) between u and x,y is not equal to the difference of distances between v and x,y. The minimum weighted drs problem consists of finding a DRS in G with minimum total weight. We establish Θ(ln n) approximability of the minimum drs problem on general graphs for both weighted and unweighted versions. This is the first work providing explicit approximation lower and upper bounds for minimum (weighted) drs problem, which are nearly tight. Moreover, we design first known strongly polynomial time algorithms for the minimum weighted drs problem on general wheels and trees with additional constant k ≥ 0 edges.

Research supported in part by NNSF of China under Grant No. 11222109, 11021161 and 10928102, by 973 Project of China under Grant No. 2011CB80800, and by CAS Program for Cross & Cooperative Team of Science & Technology Innovation.

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 39.99
Price excludes VAT (USA)
  • Available as 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

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Berman, P., DasGupta, B., Kao, M.Y.: Tight approximability results for test set problems in bioinformatics. J. Comput. Syst. Sci. 71(2), 145–162 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  2. Cáceres, J., Hernando, C., Mora, M., Pelayo, I.M., Puertas, M.L., Seara, C., Wood, D.R.: On the metric dimension of cartesian products of graphs. SIAM J. Discrete Math. 21(2), 423–441 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  3. Čangalović, M., Kratica, J., Kovačević-Vujčić, V., Stojanović, M.: Minimal doubly resolving sets of prism graphs. Optimization (ahead-of-print), 1–7 (2013)

    Google Scholar 

  4. Chartrand, G., Zhang, P.: The theory and applications of resolvability in graphs: A survey. Congressus Numerantium 160, 47–68 (2003)

    MATH  MathSciNet  Google Scholar 

  5. Chen, X., Hu, X., Wang, C.: Approximability of the minimum weighted doubly resolving set problem. CoRR abs/1404.4676 (2014)

    Google Scholar 

  6. Chlebík, M., Chlebíková, J.: Approximation hardness of dominating set problems in bounded degree graphs. Inf. Comput. 206(11), 1264–1275 (2008)

    Article  MATH  Google Scholar 

  7. Díaz, J., Pottonen, O., Serna, M., van Leeuwen, E.J.: On the complexity of metric dimension. In: Epstein, L., Ferragina, P. (eds.) ESA 2012. LNCS, vol. 7501, pp. 419–430. Springer, Heidelberg (2012)

    Chapter  Google Scholar 

  8. Epstein, L., Levin, A., Woeginger, G.J.: The (weighted) metric dimension of graphs: hard and easy cases. In: Golumbic, M.C., Stern, M., Levy, A., Morgenstern, G. (eds.) WG 2012. LNCS, vol. 7551, pp. 114–125. Springer, Heidelberg (2012)

    Chapter  Google Scholar 

  9. Gomez Rodriguez, M., Leskovec, J., Krause, A.: Inferring networks of diffusion and influence. In: Proc. of the 16th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, KDD 2010, pp. 1019–1028 (2010)

    Google Scholar 

  10. Hauptmann, M., Schmied, R., Viehmann, C.: Approximation complexity of metric dimension problem. J. Discrete Algorithms 14, 214–222 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  11. Hochbaum, D.S., Levin, A.: Optimizing over consecutive 1’s and circular 1’s constraints. SIAM J. Optimization 17(2), 311–330 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  12. Kratica, J., Kovačević-Vujčić, V., Čangalović, M., Stojanović, M.: Minimal doubly resolving sets and the strong metric dimension of hamming graphs. Appl. Anal. Discret. Math. 6(1), 63–71 (2012)

    Article  MATH  Google Scholar 

  13. Kratica, J., Kovačević-Vujčić, V., Čangalović, M., Stojanović, M.: Minimal doubly resolving sets and the strong metric dimension of some convex polytopes. Appl. Math. Comput. 218(19), 9790–9801 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  14. Kratica, J., Čangalović, M., Kovačević-Vujčić, V.: Computing minimal doubly resolving sets of graphs. Comput. Oper. Res. 36(7), 2149–2159 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  15. Mladenović, N., Kratica, J., Kovačević-Vujčić, V., Čangalović, M.: Variable neighborhood search for metric dimension and minimal doubly resolving set problems. Eur. J. Oper. Res. 220(2), 328–337 (2012)

    Article  MATH  Google Scholar 

  16. Pinto, P.C., Thiran, P., Vetterli, M.: Locating the source of diffusion in large-scale networks. Phys. Rev. Lett. 109, 068702 (2012)

    Google Scholar 

  17. Shah, D., Zaman, T.: Rumors in a network: Who’s the culprit? IEEE Trans. Information Theory 57(8), 5163–5181 (2011)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Applied Mathematics, AMSS, Chinese Academy of Sciences, Beijing, 100190, China

    Xujin Chen & Changjun Wang

Authors
  1. Xujin Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Changjun Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Department of Computer Science, Georgia State University, 34 Peachtree Street, Suite 1410, 30303, Atlanta, GA, USA

    Zhipeng Cai

  2. Department of Computer Science, Georgia State University, 34 Peachtree Street, Suite 1443, 30303, Atlanta, GA, USA

    Alex Zelikovsky

  3. Department of Computer Science, Georgia State University, 34 Peachtree Street, Suite 1449, 30303, Atlanta, GA, USA

    Anu Bourgeois

Rights and permissions

Reprints and permissions

Copyright information

© 2014 Springer International Publishing Switzerland

About this paper

Cite this paper

Chen, X., Wang, C. (2014). Approximability of the Minimum Weighted Doubly Resolving Set Problem. In: Cai, Z., Zelikovsky, A., Bourgeois, A. (eds) Computing and Combinatorics. COCOON 2014. Lecture Notes in Computer Science, vol 8591. Springer, Cham. https://doi.org/10.1007/978-3-319-08783-2_31

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-08783-2_31

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-08782-5

  • Online ISBN: 978-3-319-08783-2

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation