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Hardness of Metric Dimension in Graphs of Constant Treewidth

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Abstract

The Metric Dimension problem asks for a minimum-sized resolving set in a given (unweighted, undirected) graph G. Here, a set \(S \subseteq V(G)\) is resolving if no two distinct vertices of G have the same distance vector to S. The complexity of Metric Dimension in graphs of bounded treewidth remained elusive in the past years. Recently, Bonnet and Purohit [IPEC 2019] showed that the problem is W[1]-hard under treewidth parameterization. In this work, we strengthen their lower bound to show that Metric Dimension is NP-hard in graphs of treewidth \(24\).

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References

  1. Belmonte, R., Fomin, F.V., Golovach, P.A., Ramanujan, M.S.: Metric dimension of bounded tree-length graphs. SIAM J. Discret. Math. 31(2), 1217–1243 (2017). https://doi.org/10.1137/16M1057383

    Article  MathSciNet  MATH  Google Scholar 

  2. Bonnet, E., Purohit, N.: Metric dimension parameterized by treewidth. In Bart M. P. Jansen and Jan Arne Telle, editors, 14th International Symposium on Parameterized and Exact Computation, IPEC 2019, September 11-13, 2019, Munich, Germany, volume 148 of LIPIcs, pages 5:1–5:15. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, (2019) https://doi.org/10.4230/LIPIcs.IPEC.2019.5

  3. Díaz, J., Pottonen, O., Serna, M.J., van Leeuwen, E.J.: Complexity of metric dimension on planar graphs. J. Comput. Syst. Sci. 83(1), 132–158 (2017). https://doi.org/10.1016/j.jcss.2016.06.006

    Article  MathSciNet  MATH  Google Scholar 

  4. Eppstein, D.: Metric dimension parameterized by max leaf number. J. Graph Algorithms Appl. 19(1), 313–323 (2015). https://doi.org/10.7155/jgaa.00360

    Article  MathSciNet  MATH  Google Scholar 

  5. Epstein, L., Levin, A., Woeginger, G.J.: The (weighted) metric dimension of graphs: Hard and easy cases. Algorithmica 72(4), 1130–1171 (2015). https://doi.org/10.1007/s00453-014-9896-2

    Article  MathSciNet  MATH  Google Scholar 

  6. Fernau, H., Heggernes, P., van ’t Hof, P., Meister, D., Saei, R.: Computing the metric dimension for chain graphs. Inf. Process. Lett. 115(9), 671–676 (2015). https://doi.org/10.1016/j.ipl.2015.04.006

    Article  MathSciNet  MATH  Google Scholar 

  7. 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  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  9. 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, pages 266–276. IEEE Computer Society, (2013) https://doi.org/10.1109/CCC.2013.36

  10. Karp, R.M.: Reducibility among combinatorial problems. In: Complexity of computer computations, pages 85–103. Springer, (1972)

  11. Khuller, S., Raghavachari, B., Rosenfeld, A.: Landmarks in graphs. Discret. Appl. Math. 70(3), 217–229 (1996). https://doi.org/10.1016/0166-218X(95)00106-2

    Article  MathSciNet  MATH  Google Scholar 

  12. Kirousis, L.M., Papadimitriou, C.H.: Interval graphs and seatching. Discret. Math. 55(2), 181–184 (1985)

    Article  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Institute of Informatics, University of Warsaw, Warsaw, Poland

    Shaohua Li & Marcin Pilipczuk

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  1. Shaohua Li
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  2. Marcin Pilipczuk
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Correspondence to Shaohua Li.

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The authors of this paper, Shaohua Li and Marcin Pilipczuk, certify that there is no actual or potential conflict of interest in relation to this article.

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This research is a part of projects that have received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme under grant agreement No 714704. A preliminary version appeared at IPEC 2021

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Li, S., Pilipczuk, M. Hardness of Metric Dimension in Graphs of Constant Treewidth. Algorithmica 84, 3110–3155 (2022). https://doi.org/10.1007/s00453-022-01005-y

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