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Metric Dimension and Geodetic Set Parameterized by Vertex Cover

Authors Florent Foucaud , Esther Galby , Liana Khazaliya , Shaohua Li , Fionn Mc Inerney , Roohani Sharma , Prafullkumar Tale

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Author Details

Florent Foucaud
  • Université Clermont Auvergne, CNRS, Mines Saint-Étienne, Clermont Auvergne INP, LIMOS, 63000 Clermont-Ferrand, France
Esther Galby
  • Department of Computer Science and Engineering, Chalmers University of Technology and University of Gothenburg, Sweden
Liana Khazaliya
  • Technische Universität Wien, Austria
Shaohua Li
  • School of Computer Science and Engineering, Central South University, Changsha, China
Fionn Mc Inerney
  • Telefónica Scientific Research, Barcelona, Spain
Roohani Sharma
  • University of Bergen, Norway
Prafullkumar Tale
  • Indian Institute of Science Education and Research Pune, India

Funding

  • Foucaud, Florent: ANR project GRALMECO (ANR-21-CE48-0004), French government IDEX-ISITE initiative 16-IDEX-0001 (CAP 20-25), International Research Center "Innovation Transportation and Production Systems" of the I-SITE CAP 20-25.
  • Khazaliya, Liana: Vienna Science and Technology Fund (WWTF) [10.47379/ICT22029]; Austrian Science Fund (FWF) [10.55776/Y1329]; European Union’s Horizon 2020 COFUND programme [LogiCS@TUWien, grant agreement No. 101034440].
  • Li, Shaohua: National Natural Science Foundation of China under Grant 62472449.
  • Mc Inerney, Fionn: Smart Networks and Services Joint Undertaking (SNS JU) under the EU’s Horizon Europe and innovation programme under Grant Agreement No. 101139067 (ELASTIC).

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Florent Foucaud, Esther Galby, Liana Khazaliya, Shaohua Li, Fionn Mc Inerney, Roohani Sharma, and Prafullkumar Tale. Metric Dimension and Geodetic Set Parameterized by Vertex Cover. In 42nd International Symposium on Theoretical Aspects of Computer Science (STACS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 327, pp. 33:1-33:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.STACS.2025.33

Abstract

For a graph G, a subset S ⊆ V(G) is called a resolving set of G if, for any two vertices u,v ∈ V(G), there exists a vertex w ∈ S such that d(w,u) ≠ d(w,v). The Metric Dimension problem takes as input a graph G on n vertices and a positive integer k, and asks whether there exists a resolving set of size at most k. In another metric-based graph problem, Geodetic Set, the input is a graph G and an integer k, and the objective is to determine whether there exists a subset S ⊆ V(G) of size at most k such that, for any vertex u ∈ V(G), there are two vertices s₁, s₂ ∈ S such that u lies on a shortest path from s₁ to s₂.
These two classical problems are known to be intractable with respect to the natural parameter, i.e., the solution size, as well as most structural parameters, including the feedback vertex set number and pathwidth. We observe that both problems admit an FPT algorithm running in 2^𝒪(vc²) ⋅ n^𝒪(1) time, and a kernelization algorithm that outputs a kernel with 2^𝒪(vc) vertices, where vc is the vertex cover number. We prove that unless the Exponential Time Hypothesis (ETH) fails, Metric Dimension and Geodetic Set, even on graphs of bounded diameter, do not admit  
- an FPT algorithm running in 2^o(vc²) ⋅ n^𝒪(1) time, nor 
- a kernelization algorithm that does not increase the solution size and outputs a kernel with 2^o(vc) vertices.  We only know of one other problem in the literature that admits such a tight algorithmic lower bound with respect to vc. Similarly, the list of known problems with exponential lower bounds on the number of vertices in kernelized instances is very short.

Subject Classification

ACM Subject Classification
  • Theory of computation → Parameterized complexity and exact algorithms
Keywords
  • Parameterized Complexity
  • ETH-based Lower Bounds
  • Kernelization
  • Vertex Cover
  • Metric Dimension
  • Geodetic Set

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References

  1. A. Agrawal, D. Lokshtanov, S. Saurabh, and M. Zehavi. Split contraction: The untold story. ACM Trans. Comput. Theory, 11(3):18:1-18:22, 2019. URL: https://doi.org/10.1145/3319909.
  2. R. Belmonte, F. V. Fomin, P. A. Golovach, and M. S. Ramanujan. Metric dimension of bounded tree-length graphs. SIAM J. Discrete Math., 31(2):1217-1243, 2017. URL: https://doi.org/10.1137/16M1057383.
  3. B. Bergougnoux, O. Defrain, and F. Mc Inerney. Enumerating minimal solution sets for metric graph problems. In Proc. of the 50th International Workshop on Graph-Theoretic Concepts in Computer Science (WG 2024), volume 14760 of Lecture Notes in Computer Science, pages 50-64. Springer, 2024. Google Scholar
  4. E. Bonnet and N. Purohit. Metric dimension parameterized by treewidth. Algorithmica, 83:2606-2633, 2021. URL: https://doi.org/10.1007/S00453-021-00808-9.
  5. N. Bousquet, Q. Deschamps, and A. Parreau. Metric dimension parameterized by treewidth in chordal graphs. In Proc. of the 49th International Workshop on Graph-Theoretic Concepts in Computer Science (WG 2023), volume 14093 of Lecture Notes in Computer Science, pages 130-142. Springer, 2023. URL: https://doi.org/10.1007/978-3-031-43380-1_10.
  6. D. Chakraborty, S. Das, F. Foucaud, H. Gahlawat, D. Lajou, and B. Roy. Algorithms and complexity for geodetic sets on planar and chordal graphs. In Proc. of the 31st International Symposium on Algorithms and Computation (ISAAC 2020), volume 181 of LIPIcs, pages 7:1-7:15. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2020. URL: https://doi.org/10.4230/LIPICS.ISAAC.2020.7.
  7. D. Chakraborty, F. Foucaud, D. Majumdar, and P. Tale. Tight (double) exponential bounds for identification problems: Locating-dominating set and test cover. In Proc. of the 35th International Symposium on Algorithms and Computation (ISAAC 2024), volume 322 of LIPIcs, pages 19:1-19:18. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2024. URL: https://doi.org/10.4230/LIPICS.ISAAC.2024.19.
  8. J. Chalopin, V. Chepoi, F. Mc Inerney, and S. Ratel. Non-clashing teaching maps for balls in graphs. In Proc. of the 37th Annual Conference on Learning Theory (COLT 2024), volume 247 of Proceedings of Machine Learning Research, pages 840-875. PMLR, 2024. URL: https://proceedings.mlr.press/v247/chalopin24a.html.
  9. L. S. Chandran, D. Issac, and A. Karrenbauer. On the parameterized complexity of biclique cover and partition. In Proc. of the 11th International Symposium on Parameterized and Exact Computation (IPEC 2016), volume 63 of LIPIcs, pages 11:1-11:13. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2016. URL: https://doi.org/10.4230/LIPICS.IPEC.2016.11.
  10. G. Chartrand, F. Harary, and P. Zhang. On the geodetic number of a graph. Networks, 39(1):1-6, 2002. URL: https://doi.org/10.1002/NET.10007.
  11. J. Chen, I. A. Kanj, and W. Jia. Vertex cover: Further observations and further improvements. J. Algorithms, 41(2):280-301, 2001. URL: https://doi.org/10.1006/JAGM.2001.1186.
  12. M. Cygan, F. V. Fomin, L. Kowalik, D. Lokshtanov, D. Marx, M. Pilipczuk, M. Pilipczuk, and S. Saurabh. Parameterized Algorithms. Springer, 2015. Google Scholar
  13. M. Cygan, M. Pilipczuk, and M. Pilipczuk. Known algorithms for edge clique cover are probably optimal. SIAM J. Comput., 45(1):67-83, 2016. URL: https://doi.org/10.1137/130947076.
  14. R. Diestel. Graph Theory, 6th Edition, volume 173 of Graduate texts in mathematics. Springer, 2024. Google Scholar
  15. M. C. Dourado, F. Protti, D. Rautenbach, and J. L. Szwarcfiter. Some remarks on the geodetic number of a graph. Discrete Mathematics, 310(4):832-837, 2010. URL: https://doi.org/10.1016/J.DISC.2009.09.018.
  16. M. Dumas, F. Foucaud, A. Perez, and I. Todinca. On graphs coverable by k shortest paths. SIAM J. Discrete Math., 38(2):1840-1862, 2024. URL: https://doi.org/10.1137/23M1564511.
  17. D. Eppstein. Metric dimension parameterized by max leaf number. Journal of Graph Algorithms and Applications, 19(1):313-323, 2015. URL: https://doi.org/10.7155/JGAA.00360.
  18. L. Epstein, A. Levin, and G. J. Woeginger. The (weighted) metric dimension of graphs: Hard and easy cases. Algorithmica, 72(4):1130-1171, 2015. URL: https://doi.org/10.1007/S00453-014-9896-2.
  19. F. Foucaud, E. Galby, L. Khazaliya, S. Li, F. Mc Inerney, R. Sharma, and P. Tale. Metric dimension and geodetic set parameterized by vertex cover. CoRR, abs/2405.01344, 2024. URL: https://doi.org/10.48550/arXiv.2405.01344.
  20. F. Foucaud, E. Galby, L. Khazaliya, S. Li, F. Mc Inerney, R. Sharma, and P. Tale. Problems in NP can admit double-exponential lower bounds when parameterized by treewidth or vertex cover. In Proc. of the 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024), volume 297 of LIPIcs, pages 66:1-66:19. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2024. URL: https://doi.org/10.4230/LIPICS.ICALP.2024.66.
  21. E. Galby, L. Khazaliya, F. Mc Inerney, R. Sharma, and P. Tale. Metric dimension parameterized by feedback vertex set and other structural parameters. SIAM J. Discrete Math., 37(4):2241-2264, 2023. URL: https://doi.org/10.1137/22M1510911.
  22. M. R. Garey and D. S. Johnson. Computers and Intractability - A guide to NP-completeness. W.H. Freeman and Company, 1979. Google Scholar
  23. T. Gima, T. Hanaka, M. Kiyomi, Y. Kobayashi, and Y. Otachi. Exploring the gap between treedepth and vertex cover through vertex integrity. Theoretical Comp. Sci., 918:60-76, 2022. URL: https://doi.org/10.1016/J.TCS.2022.03.021.
  24. G. Z. Gutin, M. S. Ramanujan, F. Reidl, and M. Wahlström. Alternative parameterizations of metric dimension. Theoretical Comp. Sci., 806:133-143, 2020. URL: https://doi.org/10.1016/J.TCS.2019.01.028.
  25. F. Harary, E. Loukakis, and C. Tsouros. The geodetic number of a graph. Mathematical and Computer Modelling, 17(11):89-95, 1993. Google Scholar
  26. F. Harary and R. A. Melter. On the metric dimension of a graph. Ars Comb., 2:191-195, 1976. Google Scholar
  27. D. G. Harris and N. S. Narayanaswamy. A faster algorithm for vertex cover parameterized by solution size. In Proc. of the 41st International Symposium on Theoretical Aspects of Computer Science (STACS 2024), volume 289 of LIPIcs, pages 40:1-40:18. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2024. URL: https://doi.org/10.4230/LIPICS.STACS.2024.40.
  28. S. Hartung and A. Nichterlein. On the parameterized and approximation hardness of metric dimension. In Proc. of the 28th Conference on Computational Complexity, CCC 2013, pages 266-276. IEEE Computer Society, 2013. URL: https://doi.org/10.1109/CCC.2013.36.
  29. L. Jaffke, O.-J. Kwon, T. J. F. Strømme, and J. A. Telle. Mim-width III. Graph powers and generalized distance domination problems. Theoretical Comp. Sci., 796:216-236, 2019. URL: https://doi.org/10.1016/J.TCS.2019.09.012.
  30. I. Katsikarelis, M. Lampis, and V. Th. Paschos. Structurally parameterized d-scattered set. Discrete Applied Mathematics, 308:168-186, 2022. URL: https://doi.org/10.1016/J.DAM.2020.03.052.
  31. L. Kellerhals and T. Koana. Parameterized complexity of geodetic set. Journal of Graph Algorithms and Applications, 26(4):401-419, 2022. URL: https://doi.org/10.7155/JGAA.00601.
  32. M. Lampis, N. Melissinos, and M. Vasilakis. Parameterized max min feedback vertex set. In Proc. of the 48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023), volume 272 of LIPIcs, pages 62:1-62:15. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2023. URL: https://doi.org/10.4230/LIPICS.MFCS.2023.62.
  33. S. Li and M. Pilipczuk. Hardness of metric dimension in graphs of constant treewidth. Algorithmica, 84(11):3110-3155, 2022. URL: https://doi.org/10.1007/S00453-022-01005-Y.
  34. C. V. G. C. Lima, V. F. dos Santos, J. H. G. Sousa, and S. A. Urrutia. On the computational complexity of the strong geodetic recognition problem. RAIRO Oper. Res., 58(5):3755-3770, 2024. URL: https://doi.org/10.1051/RO/2024120.
  35. D. Marx, M. Pilipczuk, and M. Pilipczuk. On subexponential parameterized algorithms for steiner tree and directed subset TSP on planar graphs. In Proc. of the 59th IEEE Annual Symposium on Foundations of Computer Science (FOCS 2018), pages 474-484, 2018. Google Scholar
  36. M. Pilipczuk. Problems parameterized by treewidth tractable in single exponential time: A logical approach. In Proc. of the 36th International Symposium on Mathematical Foundations of Computer Science (MFCS 2011), volume 6907 of Lecture Notes in Computer Science, pages 520-531. Springer, 2011. URL: https://doi.org/10.1007/978-3-642-22993-0_47.
  37. I. Sau and U. dos Santos Souza. Hitting forbidden induced subgraphs on bounded treewidth graphs. Inf. Comput., 281:104812, 2021. URL: https://doi.org/10.1016/J.IC.2021.104812.
  38. P. J. Slater. Leaves of trees. In Proc. of the 6th Southeastern Conf. on Combinatorics, Graph Theory, and Computing, pages 549-559. Congressus Numerantium, No. XIV. Util. Math., 1975. Google Scholar
  39. P. Tale. Double exponential lower bound for telephone broadcast. CoRR, abs/2403.03501, 2024. URL: https://doi.org/10.48550/arXiv.2403.03501.
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