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The Complexity of the Identifying Code Problem in Restricted Graph Classes

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Combinatorial Algorithms (IWOCA 2013)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 8288))

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Abstract

An identifying code is a subset of vertices of a graph such that each vertex is uniquely determined by its nonempty neighbourhood within the identifying code. We study the associated computational problem Minimum Identifying Code, which is known to be NP-hard, even when the input graph belongs to a number of specific graph classes such as planar bipartite graphs. Though the problem is approximable within a logarithmic factor, it is known to be hard to approximate within any sub-logarithmic factor. We extend the latter result to the case where the input graph is bipartite, split or co-bipartite. Among other results, we also show that for bipartite graphs of bounded maximum degree (at least 3), it is hard to approximate within some constant factor. We summarize known results in the area, and we compare them to the ones for the related problem Minimum Dominating Set. In particular, our work exhibits important graph classes for which Minimum Dominating Set is efficiently solvable, but Minimum Identifying Code is hard (whereas in all previously studied classes, their complexity is the same). We also introduce a graph class for which the converse holds.

An extended version of this paper, containing the full proofs and further results, is available on the author’s website [16].

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References

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

    Article  MathSciNet  MATH  Google Scholar 

  2. Auger, D., Charon, I., Hudry, O., Lobstein, A.: Complexity results for identifying codes in planar graphs. Int. T. Oper. Res. 17(6), 691–710 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  3. Ausiello, G., Crescenzi, P., Gambosi, G., Kann, V., Marchetti-Spaccamela, A., Protasi, M.: Complexity and approximation. Springer (1999)

    Google Scholar 

  4. Baker, B.S.: Approximation algorithms for NP-complete problems on planar graphs. J. ACM 41(1), 153–180 (1994)

    Article  MATH  Google Scholar 

  5. Berger-Wolf, T.Y., Laifenfeld, M., Trachtenberg, A.: Identifying codes and the set cover problem. In: Proc. 44th Allerton Conf. on Comm. Contr. and Comput. (2006)

    Google Scholar 

  6. Booth, K.S., Johnson, H.J.: Dominating sets in chordal graphs. SIAM J. Comput. 11(1), 191–199 (1982)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  8. Clark, B.N., Colbourn, C.J., Johnson, D.S.: Unit disk graphs. Discrete Math. 86(1-3), 165–177 (1990)

    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. Chlebík, M., Chlebíková, J.: Approximation hardness of edge dominating set problems. J. Comb. Optim. 11, 279–290 (2006)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  12. De Bontridder, K.M.J., Halldórsson, B.V., Halldórsson, M.M., Hurkens, C.A.J., Lenstra, J.K., Ravi, R., Stougie, L.: Approximation algorithms for the test cover problem. Math. Programm. Ser. B 98, 477–491 (2003)

    Article  MATH  Google Scholar 

  13. De Ridder, H.N., et al.: Information System on Graph Classes and their Inclusions (ISGCI), http://www.graphclasses.org

  14. Farber, M., Keil, J.M.: Domination in permutation graphs. J. Algorithm. 6, 309–321 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  15. Foucaud, F.: Combinatorial and algorithmic aspects of identifying codes in graphs. PhD thesis, Université Bordeaux 1, France (2012), http://tel.archives-ouvertes.fr/tel-00766138

  16. Foucaud, F.: On the decision and approximation complexities for identifying codes and locating-dominating sets in restricted graph classes (2013) (manuscript), http://www-ma4.upc.edu/~florent.foucaud/Research

  17. Foucaud, F., Gravier, S., Naserasr, R., Parreau, A., Valicov, P.: Identifying codes in line graphs. J. Graph Theor. 73(4), 425–448 (2013), doi:10.1002/jgt.21686

    Article  MathSciNet  MATH  Google Scholar 

  18. Foucaud, F., Mertzios, G., Naserasr, R., Parreau, A., Valicov, P.: Identifying codes in subclasses of perfect graphs. Manuscript (2012)

    Google Scholar 

  19. Garey, M.R., Johnson, D.S.: The rectilinear Steiner tree problem is NP-complete. SIAM J. Appl. Math. 32(4), 826–834 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  20. Garey, M.R., Johnson, D.S.: Computers and intractability: a guide to the theory of NP-completeness. W. H. Freeman (1979)

    Google Scholar 

  21. Gravier, S., Klasing, R., Moncel, J.: Hardness results and approximation algorithms for identifying codes and locating-dominating codes in graphs. Alg. Oper. Res. 3(1), 43–50 (2008)

    MathSciNet  MATH  Google Scholar 

  22. Haynes, T.W., Hedetniemi, S.T., Slater, P.J. (eds.): Domination in graphs: advanced topics. Marcel Dekker (1998)

    Google Scholar 

  23. Hunt III, H.B., Marathe, M.V., Radhakrishnan, V., Ravi, S.S., Rosenkrantz, D.J., Stearns, R.E.: NC-approximation schemes for NP-and PSPACE-hard problems for geometric graphs. J. Algor. 26, 238–274 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  24. Johnson, D.S.: Approximation algorithms for combinatorial problems. J. Comput. Sys. Sci. 9, 256–278 (1974)

    Article  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  26. Laifenfeld, M., Trachtenberg, A.: Identifying codes and covering problems. IEEE T. Inform. Theory 54(9), 3929–3950 (2008)

    Article  MathSciNet  Google Scholar 

  27. Lobstein, A.: Watching systems, identifying, locating-dominating and discriminating codes in graphs: a bibliography, http://www.infres.enst.fr/~lobstein/debutBIBidetlocdom.pdf

  28. Müller, H., Brandtädt, A.: The NP-completeness of Steiner Tree and Dominating Set for chordal bipartite graphs. Theor. Comput. Sci. 53, 257–265 (1987)

    Article  MATH  Google Scholar 

  29. Müller, T., Sereni, J.-S.: Identifying and locating-dominating codes in (random) geometric networks. Comb. Probab. Comput. 18(6), 925–952 (2009)

    Article  MATH  Google Scholar 

  30. Papadimitriou, C.H., Yannakakis, M.: Optimization, approximation, and complexity classes. J. Comput. Sys. Sci. 43(3), 425–440 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  31. Slater, P.J., Rall, D.F.: On location-domination numbers for certain classes of graphs. Congr. Numer. 45, 97–106 (1984)

    MathSciNet  Google Scholar 

  32. Suomela, J.: Approximability of identifying codes and locating-dominating codes. Inform. Process. Lett. 103(1), 28–33 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  33. Suomela, J.: Answer to the question “Is the dominating set problem restricted to planar bipartite graphs of maximum degree 3 NP-complete?”, http://cstheory.stackexchange.com/a/2508/1930

  34. Ungrangsi, R., Trachtenberg, A., Starobinski, D.: An implementation of indoor location detection systems based on identifying codes. In: Aagesen, F.A., Anutariya, C., Wuwongse, V. (eds.) INTELLCOMM 2004. LNCS, vol. 3283, pp. 175–189. Springer, Heidelberg (2004)

    Chapter  Google Scholar 

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Authors and Affiliations

  1. Universitat Politècnica de Catalunya, Building C3, C/ Jordi Girona 1-3, 08034, Barcelona, Spain

    Florent Foucaud

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  1. Florent Foucaud
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Editors and Affiliations

  1. UFR des Sciences et Techniques, LITIS, Université de Rouen, 76821, EA 4180, Mont Saint Aignan Cedex, France

    Thierry Lecroq  & Laurent Mouchard  & 

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Foucaud, F. (2013). The Complexity of the Identifying Code Problem in Restricted Graph Classes. In: Lecroq, T., Mouchard, L. (eds) Combinatorial Algorithms. IWOCA 2013. Lecture Notes in Computer Science, vol 8288. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-45278-9_14

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  • DOI: https://doi.org/10.1007/978-3-642-45278-9_14

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-45277-2

  • Online ISBN: 978-3-642-45278-9

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