Skip to main content
SpringerLink
Log in

On the Distance Identifying Set Meta-problem and Applications to the Complexity of Identifying Problems on Graphs

  • Published:
Algorithmica Aims and scope Submit manuscript

Abstract

Numerous problems consisting in identifying vertices in graphs using distances are useful in domains such as network verification and graph isomorphism. Unifying them into a meta-problem may be of main interest. We introduce here a promising solution named Distance Identifying Set. The model contains Identifying Code (IC), Locating Dominating Set (LD) and their generalizations r-IC and r-LD where the closed neighborhood is considered up to distance r. It also contains Metric Dimension (MD) and its refinement r-MD in which the distance between two vertices is considered as infinite if the real distance exceeds r. Note that while IC = 1-IC and LD = 1-LD, we have MD = \(\infty\)-MD; we say that MD is not local. In this article, we prove computational lower bounds for several problems included in Distance Identifying Set by providing generic reductions from (Planar) Hitting Set to the meta-problem. We focus on two families of problems from the meta-problem: the first one, called local, contains r-IC, r-LD and r-MD for each positive integer r while the second one, called 1-layered, contains LD, MD and r-MD for each positive integer r. We have: (1) the 1-layered problems are NP-hard even in bipartite apex graphs, (2) the local problems are NP-hard even in bipartite planar graphs, (3) assuming ETH, all these problems cannot be solved in \(2^{o(\sqrt{n})}\) when restricted to bipartite planar or apex graph, respectively, and they cannot be solved in \(2^{o(n)}\) on bipartite graphs, and (4) except if \({\mathsf{W}[0]} = {\mathsf{W}[2]}\), they do not admit parameterized algorithms in \(2^{{\mathcal {O}}(k)} \cdot n^{{\mathcal {O}}(1)}\) even when restricted to bipartite graphs. Here k is the solution size of a relevant identifying set. In particular, Metric Dimension cannot be solved in \(2^{o(n)}\) under ETH, answering a question of Hartung and Nichterlein (Proceedings of the 28th conference on computational complexity, CCC, 2013).

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
Fig. 2
Fig. 3
Fig. 4
Fig. 5

Similar content being viewed by others

References

  1. Auger, D.: Minimal identifying codes in trees and planar graphs with large girth. Eur. J. Comb. 31(5), 1372–1384 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  2. Babai, L.: On the complexity of canonical labeling of strongly regular graphs. SIAM J. Comput. 9(1), 212–216 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bampas, E., Bilò, D., Drovandi, G., Gualà, L., Klasing, R., Proietti, G.: Network verification via routing table queries. J. Comput. Syst. Sci. 81(1), 234–248 (2015)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  5. Belmonte, R., Fomin, F.V., Golovach, P.A., Ramanujan, M.S.: Metric dimension of bounded tree-length graphs. SIAM J. Discrete Math. 31, 1217–1243 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bollobás, B., Scott, A.D.: On separating systems. Eur. J. Comb. 28(4), 1068–1071 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  7. Bondy, J.A.: Induced subsets. J. Combi. Theory Ser. B 12(2), 201–202 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  8. Charbit, E., Charon, I., Cohen, G.D., Hudry, O., Lobstein, A.: Discriminating codes in bipartite graphs: bounds, extremal cardinalities, complexity. Adv. Math. Commun. 2(4), 403–420 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  9. Charon, I., Hudry, O., Lobstein, A.: Minimizing the size of an identifying or locating-dominating code in a graph is np-hard. Theor. Comput. Sci. 290(3), 2109–2120 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  10. Cohen, G.D., Honkala, I.S., Lobstein, A., Zémor, G.: On identifying codes. In: Codes and Association Schemes, Proceedings of a DIMACS Workshop, Piscataway, New Jersey, USA, 9–12 November, 1999, pp. 97–110 (1999)

  11. Colbourn, C.J., Slater, P.J., Stewart, L.K.: Locating dominating sets in series parallel networks. Congr. Numer. 56, 135–162 (1987)

    MathSciNet  MATH  Google Scholar 

  12. Díaz, J., Pottonen, O., Serna, M.J., Jan van Leeuwen, E.: On the complexity of metric dimension. In: Algorithms—ESA 2012—20th Annual European Symposium, Ljubljana, Slovenia, 10–12 September, 2012. Proceedings, pp. 419–430 (2012)

  13. Downey, R.G., Fellows, M.R.: Parameterized Complexity. Monographs in Computer Science. Springer, Berlin (1999)

    Book  Google Scholar 

  14. Epstein, L., Levin, A., Woeginger, G.J.: The (weighted) metric dimension of graphs: Hard and easy cases. Algorithmica 72(4), 1130–1171 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  15. Foucaud, F., Mertzios, G.B., Naserasr, R., Parreau, A., Valicov, P.: Identification, location-domination and metric dimension on interval and permutation graphs. II. Algorithms and complexity. Algorithmica 78(3), 914–944 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  16. Fernau, H., Heggernes, P., Hof, P., Meister, D., Saei, R.: Computing the metric dimension for chain graphs. Inf. Process. Lett. 115(9), 671–676 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  17. Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman, New York (1979)

    MATH  Google Scholar 

  18. Harary, F., Melter, R.A.: On the metric dimension of a graph. Ars Combin. 2(191–195), 1 (1976)

    MATH  Google Scholar 

  19. Hartung, S.: Exploring parameter spaces in coping with computational intractability. Ph.D. thesis, Fakultät Elektrotechnik und Informatik der Technischen Universität Berlin, 12 (2014)

  20. Hartung, S., Nichterlein, A.: On the parameterized and approximation hardness of metric dimension. In: Proceedings of the 28th Conference on Computational Complexity, CCC 2013, K.lo Alto, California, USA, 5–7 June, 2013, pp. 266–276 (2013)

  21. Hoffmann, S., Wanke, E.: Metric dimension for Gabriel unit disk graphs is NP-complete. In: Algorithms for Sensor Systems, 8th International Symposium on Algorithms for Sensor Systems, Wireless Ad Hoc Networks and Autonomous Mobile Entities, ALGOSENSORS 2012, Ljubljana, Slovenia, 13–14 September 2012. Revised Selected Papers, pp. 90–92 (2012)

  22. Hoffmann, S., Elterman, A., Wanke, E.: A linear time algorithm for metric dimension of cactus block graphs. Theor. Comput. Sci. 630, 43–62 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  23. Impagliazzo, R., Paturi, R., Zane, F.: Which problems have strongly exponential complexity? J. Comput. Syst. Sci. 63(4), 512–530 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  24. Karpovsky, M.G., Chakrabarty, K., Levitin, L.B.: On a new class of codes for identifying vertices in graphs. IEEE Trans. Inf. Theory 44(2), 599–611 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  25. Kim, J.H., Pikhurko, O., Spencer, J.H., Verbitsky, O.: How complex are random graphs in first order logic? Random Struct. Algorithms 26(1–2), 119–145 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  26. Slater, P.J.: Leaves of trees. Congr. Numer 14(549–559), 37 (1975)

    MATH  Google Scholar 

  27. Slater, P.J.: Domination and location in acyclic graphs. Networks 17(1), 55–64 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  28. Slater, P.J.: Dominating and reference sets in a graph. J. Math. Phys. Sci 22(4), 445–455 (1988)

    MathSciNet  MATH  Google Scholar 

  29. Tippenhauer, S.: On Planar 3-SAT and Its Variants. Ph.D. thesis, Freien Universität Berlin (2016)

  30. Ungrangsi, R., Trachtenberg, A., Starobinski, D.: An implementation of indoor location detection systems based on identifying codes. In: Intelligence in Communication Systems, IFIP International Conference, INTELLCOMM 2004, Bangkok, Thailand, 23–26 November 2004, Proceedings, pp. 175–189 (2004)

Download references

Acknowledgements

We want to thank Stéphane Bessy and Anaël Grandjean for their precious advice concerning the correctness of the claims, and the structure of the article.

Author information

Authors and Affiliations

  1. LIRMM, Université de Montpellier, 161 rue Ada, 34095, Montpellier, France

    Florian Barbero, Lucas Isenmann & Jocelyn Thiebaut

Authors
  1. Florian Barbero
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Lucas Isenmann
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Jocelyn Thiebaut
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Lucas Isenmann.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Barbero, F., Isenmann, L. & Thiebaut, J. On the Distance Identifying Set Meta-problem and Applications to the Complexity of Identifying Problems on Graphs. Algorithmica 82, 2243–2266 (2020). https://doi.org/10.1007/s00453-020-00674-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00453-020-00674-x

Keywords

Search

Navigation