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Resolving Sets in Temporal Graphs

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Combinatorial Algorithms (IWOCA 2024)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 14764))

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Abstract

A resolving set R in a graph G is a set of vertices such that every vertex of G is uniquely identified by its distances to the vertices of R. Introduced in the 1970’s, this concept has been since then extensively studied from both combinatorial and algorithmic point of view. We propose a generalization of the concept of resolving sets to temporal graphs, i.e., graphs with edge sets that change over discrete time-steps. In this setting, the temporal distance from u to v is the earliest possible time-step at which a journey with strictly increasing time-steps on edges leaving u reaches v, i.e., the first time-step at which v could receive a message broadcast from u. A temporal resolving set of a temporal graph \(\mathcal {G}\) is a subset R of its vertices such that every vertex of \(\mathcal {G}\) is uniquely identified by its temporal distances from vertices of R. We study the problem of finding a minimum-size temporal resolving set, and show that it is NP-complete even on very restricted graph classes and with strong constraints on the time-steps: temporal complete graphs where every edge appears in either time-step 1 or 2, temporal trees where every edge appears in at most two consecutive time-steps, and even temporal subdivided stars where every edge appears in at most two (not necessarily consecutive) time-steps. On the other hand, we give polynomial-time algorithms for temporal paths and temporal stars where every edge appears in exactly one time-step, and give a combinatorial analysis and algorithms for several temporal graph classes where the edges appear in periodic time-steps.

This work was supported by the International Research Center “Innovation Transportation and Production Systems” of the I-SITE CAP 20-25 and by the ANR project GRALMECO (ANR-21-CE48-0004). Jan Bok was also funded by the European Union (ERC, POCOCOP, 101071674). Tuomo Lehtilä was also supported by Business Finland Project 6GNTF, funding decision 10769/31/2022.

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Notes

  1. 1.

    Those are sometimes called strict journeys in the literature, but as argued in [30], strict journeys are naturally suited to applications where one cannot traverse multiple edges at the same time.

  2. 2.

    Also called a k-truncated dominating resolving set in [20].

References

  1. Arrighi, E., Grüttemeier, N., Morawietz, N., Sommer, F., Wolf, P.: Multi-parameter analysis of finding minors and subgraphs in edge-periodic temporal graphs. In: Gąsieniec, L. (ed.) SOFSEM 2023. LNCS, vol. 13878, pp. 283–297. Springer, Cham (2023). https://doi.org/10.1007/978-3-031-23101-8_19

    Chapter  Google Scholar 

  2. Bartha, Z., Komjáthy, J., Raes, J.: Sharp bound on the truncated metric dimension of trees. Discrete Math. 346(8), 113410 (2023)

    Article  MathSciNet  Google Scholar 

  3. Bellitto, T., Conchon-Kerjan, C., Escoffier, B.: Restless exploration of periodic temporal graphs. In: 2nd Symposium on Algorithmic Foundations of Dynamic Networks (SAND 2023). Schloss Dagstuhl-Leibniz-Zentrum für Informatik (2023)

    Google Scholar 

  4. Bok, J., Dailly, A., Lehtilä, T.: Resolving sets in temporal graphs. ar**v preprint ar**v:2403.13183 (2024)

  5. Casteigts, A., Flocchini, P., Quattrociocchi, W., Santoro, N.: Time-varying graphs and dynamic networks. Int. J. Parallel Emergent Distrib. Syst. 27(5), 387–408 (2012)

    Article  Google Scholar 

  6. Chartrand, G., Eroh, L., Johnson, M.A., Oellermann, O.R.: Resolvability in graphs and the metric dimension of a graph. Discrete Appl. Math. 105(1–3), 99–113 (2000)

    Article  MathSciNet  Google Scholar 

  7. De Carufel, J.L., Flocchini, P., Santoro, N., Simard, F.: Cops & robber on periodic temporal graphs: characterization and improved bounds. In: Rajsbaum, S., Balliu, A., Daymude, J.J., Olivetti, D. (eds.) SIROCCO 2023. LNCS, vol. 13892, pp. 386–405. Springer, Cham (2023). https://doi.org/10.1007/978-3-031-32733-9_17

    Chapter  Google Scholar 

  8. Díaz, J., Pottonen, O., Serna, M., van Leeuwen, E.J.: Complexity of metric dimension on planar graphs. J. Comput. Syst. Sci. 83(1), 132–158 (2017)

    Article  MathSciNet  Google Scholar 

  9. 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  Google Scholar 

  10. Erlebach, T., Spooner, J.T.: A game of cops and robbers on graphs with periodic edge-connectivity. In: Chatzigeorgiou, A., et al. (eds.) SOFSEM 2020. LNCS, vol. 12011, pp. 64–75. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-38919-2_6

    Chapter  Google Scholar 

  11. Estrada-Moreno, A., Yero, I.G., Rodríguez-Velázquez, J.A.: On the \((k, t)\)-metric dimension of graphs. Comput. J. 64(5), 707–720 (2021)

    Article  MathSciNet  Google Scholar 

  12. Fernau, H., Rodríguez-Velázquez, J.A.: On the (adjacency) metric dimension of corona and strong product graphs and their local variants: combinatorial and computational results. Discrete Appl. Math. 236, 183–202 (2018)

    Article  MathSciNet  Google Scholar 

  13. Flocchini, P., Kellett, M., Mason, P.C., Santoro, N.: Searching for black holes in subways. Theory Comput. Syst. 50, 158–184 (2012)

    Article  MathSciNet  Google Scholar 

  14. Flocchini, P., Mans, B., Santoro, N.: On the exploration of time-varying networks. Theor. Comput. Sci. 469, 53–68 (2013)

    Article  MathSciNet  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, 914–944 (2017)

    Article  MathSciNet  Google Scholar 

  16. Frongillo, R.M., Geneson, J., Lladser, M.E., Tillquist, R.C., Yi, E.: Truncated metric dimension for finite graphs. Discrete Appl. Math. 320, 150–169 (2022)

    Article  MathSciNet  Google Scholar 

  17. Galby, E., Khazaliya, L., Mc Inerney, F., Sharma, R., Tale, P.: Metric dimension parameterized by feedback vertex set and other structural parameters. SIAM J. Discrete Math. 37(4), 2241–2264 (2023)

    Article  MathSciNet  Google Scholar 

  18. Geneson, J., Yi, E.: The distance-\(k\) dimension of graphs. ar**v preprint ar**v:2106.08303 (2021)

  19. Geneson, J., Yi, E.: Broadcast dimension of graphs. Australas. J. Comb. 83, 243 (2022)

    MathSciNet  Google Scholar 

  20. Gutkovich, P., Yeoh, Z.S.: Computing truncated metric dimension of trees. ar**v preprint ar**v:2302.05960 (2023)

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

    Google Scholar 

  22. Hartung, S., Nichterlein, A.: On the parameterized and approximation hardness of metric dimension. In: 2013 IEEE Conference on Computational Complexity, pp. 266–276. IEEE (2013)

    Google Scholar 

  23. Holme, P.: Modern temporal network theory: a colloquium. Eur. Phys. J. B 88, 1–30 (2015)

    Article  Google Scholar 

  24. Holme, P., Saramäki, J.: Temporal Network Theory. Computational Social Sciences, Springer, Cham (2019)

    Book  Google Scholar 

  25. Ilcinkas, D., Wade, A.M.: On the power of waiting when exploring public transportation systems. In: Fernàndez Anta, A., Lipari, G., Roy, M. (eds.) OPODIS 2011. LNCS, vol. 7109, pp. 451–464. Springer, Heidelberg (2011). https://doi.org/10.1007/978-3-642-25873-2_31

    Chapter  Google Scholar 

  26. Jannesari, M., Omoomi, B.: The metric dimension of the lexicographic product of graphs. Discrete Math. 312(22), 3349–3356 (2012)

    Article  MathSciNet  Google Scholar 

  27. Karp, R.M.: Reducibility among combinatorial problems. In: Jünger, M., Liebling, T.M., Naddef, D., Nemhauser, G.L., Pulleyblank, W.R., Reinelt, G., Rinaldi, G., Wolsey, L.A. (eds.) 50 Years of Integer Programming 1958-2008, pp. 219–241. Springer, Heidelberg (2010). https://doi.org/10.1007/978-3-540-68279-0_8

    Chapter  Google Scholar 

  28. Kempe, D., Kleinberg, J., Kumar, A.: Connectivity and inference problems for temporal networks. In: Proceedings of the Thirty-Second Annual ACM Symposium on Theory of Computing (STOC 2000), pp. 504–513 (2000)

    Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  30. Kunz, P., Molter, H., Zehavi, M.: In which graph structures can we efficiently find temporally disjoint paths and walks? In: Elkind, E. (ed.) Proceedings of the Thirty-Second International Joint Conference on Artificial Intelligence, IJCAI-2023, pp. 180–188. International Joint Conferences on Artificial Intelligence Organization (2023)

    Google Scholar 

  31. Kuziak, D., Yero, I.G.: Metric dimension related parameters in graphs: a survey on combinatorial, computational and applied results. ar**v preprint ar**v:2107.04877 (2021)

  32. Liu, C., Wu, J.: Scalable routing in cyclic mobile networks. IEEE Trans. Parallel Distrib. Syst. 20(9), 1325–1338 (2008)

    Google Scholar 

  33. Michail, O.: An introduction to temporal graphs: an algorithmic perspective. Internet Math. 12(4), 239–280 (2016)

    Article  MathSciNet  Google Scholar 

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

    MathSciNet  Google Scholar 

  35. Tillquist, R.C., Frongillo, R.M., Lladser, M.E.: Getting the lay of the land in discrete space: a survey of metric dimension and its applications. SIAM Rev. 65(4), 919–962 (2023)

    Article  MathSciNet  Google Scholar 

  36. Zschoche, P., Fluschnik, T., Molter, H., Niedermeier, R.: The complexity of finding small separators in temporal graphs. J. Comput. Syst. Sci. 107, 72–92 (2020)

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Université Clermont-Auvergne, CNRS, Mines de Saint-Étienne, Clermont-Auvergne-INP, LIMOS, 63000, Clermont-Ferrand, France

    Jan Bok, Antoine Dailly & Tuomo Lehtilä

  2. Department of Algebra, Faculty of Mathematics and Physics, Charles University, Prague, Czechia

    Jan Bok

  3. Department of Computer Science, University of Helsinki, Helsinki, Finland

    Tuomo Lehtilä

  4. Helsinki Institute for Information Technology (HIIT), Espoo, Finland

    Tuomo Lehtilä

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Correspondence to Tuomo Lehtilä .

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  1. University of Salerno, Fisciano, Italy

    Adele Anna Rescigno

  2. University of Salerno, Fisciano, Salerno, Italy

    Ugo Vaccaro

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Bok, J., Dailly, A., Lehtilä, T. (2024). Resolving Sets in Temporal Graphs. In: Rescigno, A.A., Vaccaro, U. (eds) Combinatorial Algorithms. IWOCA 2024. Lecture Notes in Computer Science, vol 14764. Springer, Cham. https://doi.org/10.1007/978-3-031-63021-7_22

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