Skip to main content
Log in

An IP-based swapping algorithm for the metric dimension and minimal doubly resolving set problems in hypercubes

  • Original Paper
  • Published:
Optimization Letters Aims and scope Submit manuscript

Abstract

We consider the problems of determining the metric dimension and the minimum cardinality of doubly resolving sets in n-cubes. Most heuristics developed for these two NP-hard problems use a function that counts the number of pairs of vertices that are not (doubly) resolved by a given subset of vertices, which requires an exponential number of distance evaluations, with respect to n. We show that it is possible to determine whether a set of vertices (doubly) resolves the n-cube by solving an integer program with O(n) variables and O(n) constraints. We then demonstrate that small resolving and doubly resolving sets can easily be determined by solving a series of such integer programs within a swapping algorithm. Results are given for hypercubes having up to a quarter of a billion vertices, and new upper bounds are reported.

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

Access this article

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

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Beerliova, Z., Eberhard, F., Erlebach, T., Hall, A., Hoffmann, M., Mihalak, M., Ram, L.: Network discovery and verification. IEEE J. Sel. Areas Commun. 24(12), 2168–2181 (2006)

    Article  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, 423–441 (2007)

    Article  MathSciNet  Google Scholar 

  3. Cangalovic̀, M.: Private communication (2013)

  4. Chartrand, G., Eroha, L., Johnson, M., Oellermann, O.: Resolvability in graphs and the metric dimension of a graph. Discrete Appl. Math. 105, 99–113 (2000)

    Article  MathSciNet  Google Scholar 

  5. Currie, J., Oellerman, O.: The metric dimension and metric independence of a graph. J. Comb. Math. Comb. Comput. 39, 157–167 (2001)

    MathSciNet  MATH  Google Scholar 

  6. Harary, F., Melter, R.: On the metric dimension of a graph. Ars Comb. 2, 191–195 (1976)

    MATH  Google Scholar 

  7. Khuller, S., Raghavachari, B., Rosenfeld, A.: Landmarks in graphs. Discrete Appl. Math. 70(3), 217–229 (1996)

    Article  MathSciNet  Google Scholar 

  8. Kratica, J., Kovac̆evic̀-Vujc̆ic̀, V., C̆angalovic̀, M.: Computing the metric dimension of graphs by genetic algorithms. Comput. Optim. Appl. 44, 343–361 (2009)

    Article  MathSciNet  Google Scholar 

  9. Kratica, J., C̆angalovic̀, M., Kovac̆evic̀-Vujc̆ic̀, V.: Computing minimal doubly resolving sets of graphs. Comput. Oper. Res. 36, 2149–2159 (2009)

    Article  MathSciNet  Google Scholar 

  10. Mladenovic̀, N., Kratica, J., Kovac̆evic̀-Vujc̆ic̀, V., C̆angalovic̀, M.: Variable neighborhood search for metric dimension and minimal doubly resolving set problems. Eur. J. Oper. Res. 220, 328–337 (2012)

    Article  MathSciNet  Google Scholar 

  11. Murdiansyah, D., Adiwijaya, A.: Computing the metric dimension of hypercube graphs by particle swarm optimization algorithms. In: Herawan, T., Ghazali, R., Nawi, N., Deris, M. (eds.) Recent Advances on Soft Computing and Data Mining: Proceedings of the 2nd International Conference on Soft Computing and Data Mining, Bandung, Indonesia, pp. 171–178. Springer International Publishing (2017)

  12. Nikolic̀ N., C̆angalovic̀ M., Grujic̆ic̀ I.: Symmetry properties of resolving sets and metric bases in hypercubes. Optim. Lett. (2014). doi:10.1007/s11590-014-0790-2

  13. Slater, P.: Leaves of trees. Congr. Numer. 14, 549–559 (1975)

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The author would like to thank Issoufou Abdou Amadou and Gilles Éric Zagré for initiating the idea of solving an integer program to determine if a set of vertices (doubly) resolves the n-cube. Special thanks go to Serge Bisaillon for his invaluable help with the use of CPLEX.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Alain Hertz.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Hertz, A. An IP-based swapping algorithm for the metric dimension and minimal doubly resolving set problems in hypercubes. Optim Lett 14, 355–367 (2020). https://doi.org/10.1007/s11590-017-1184-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11590-017-1184-z

Keywords

Navigation