Exploiting Automorphisms of Temporal Graphs for Fast Exploration and Rendezvous

Authors Konstantinos Dogeas , Thomas Erlebach , Frank Kammer , Johannes Meintrup , William K. Moses Jr.



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Author Details

Konstantinos Dogeas
  • Department of Computer Science, Durham University, UK
Thomas Erlebach
  • Department of Computer Science, Durham University, UK
Frank Kammer
  • THM, University of Applied Sciences Mittelhessen, Gießen, Germany
Johannes Meintrup
  • HM, University of Applied Sciences Mittelhessen, Gießen, Germany
William K. Moses Jr.
  • Department of Computer Science, Durham University, UK

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Konstantinos Dogeas, Thomas Erlebach, Frank Kammer, Johannes Meintrup, and William K. Moses Jr.. Exploiting Automorphisms of Temporal Graphs for Fast Exploration and Rendezvous. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 55:1-55:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.ICALP.2024.55

Abstract

Temporal graphs are dynamic graphs where the edge set can change in each time step, while the vertex set stays the same. Exploration of temporal graphs whose snapshot in each time step is a connected graph, called connected temporal graphs, has been widely studied. In this paper, we extend the concept of graph automorphisms from static graphs to temporal graphs and show for the first time that symmetries enable faster exploration: We prove that a connected temporal graph with n vertices and orbit number r (i.e., r is the number of automorphism orbits) can be explored in O(r n^{1+ε}) time steps, for any fixed ε > 0. For r = O(n^c) for constant c < 1, this is a significant improvement over the known tight worst-case bound of Θ(n²) time steps for arbitrary connected temporal graphs. We also give two lower bounds for temporal exploration, showing that Ω(n log n) time steps are required for some inputs with r = O(1) and that Ω(rn) time steps are required for some inputs for any r with 1 ≤ r ≤ n. Moreover, we show that the techniques we develop for fast exploration can be used to derive the following result for rendezvous: Two agents with different programs and without communication ability are placed by an adversary at arbitrary vertices and given full information about the connected temporal graph, except that they do not have consistent vertex labels. Then the two agents can meet at a common vertex after O(n^{1+ε}) time steps, for any constant ε > 0. For some connected temporal graphs with the orbit number being a constant, we also present a complementary lower bound of Ω(nlog n) time steps.

Subject Classification

ACM Subject Classification
  • Theory of computation → Dynamic graph algorithms
  • Theory of computation → Graph algorithms analysis
Keywords
  • dynamic graphs
  • parameterized algorithms
  • algorithmic graph theory
  • graph automorphism
  • orbit number

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References

  1. Eric Aaron, Danny Krizanc, and Elliot Meyerson. Dmvp: foremost waypoint coverage of time-varying graphs. In International Workshop on Graph-Theoretic Concepts in Computer Science, pages 29-41. Springer, 2014. URL: https://doi.org/10.1007/978-3-319-12340-0_3.
  2. Eric Aaron, Danny Krizanc, and Elliot Meyerson. Multi-robot foremost coverage of time-varying graphs. In International Symposium on Algorithms and Experiments for Sensor Systems, Wireless Networks and Distributed Robotics, pages 22-38. Springer, 2014. URL: https://doi.org/10.1007/978-3-662-46018-4_2.
  3. Duncan Adamson, Vladimir V. Gusev, Dmitriy Malyshev, and Viktor Zamaraev. Faster exploration of some temporal graphs. In 1st Symposium on Algorithmic Foundations of Dynamic Networks (SAND 2022). Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2022. URL: https://doi.org/10.4230/LIPIcs.SAND.2022.5.
  4. Eleni C. Akrida, Jurek Czyzowicz, Leszek Gąsieniec, Łukasz Kuszner, and Paul G. Spirakis. Temporal flows in temporal networks. Journal of Computer and System Sciences, 103:46-60, 2019. URL: https://doi.org/10.1016/j.jcss.2019.02.003.
  5. Eleni C. Akrida, George B. Mertzios, Paul G. Spirakis, and Christoforos Raptopoulos. The temporal explorer who returns to the base. Journal of Computer and System Sciences, 120:179-193, 2021. URL: https://doi.org/10.1016/j.jcss.2021.04.001.
  6. Eleni C. Akrida, George B. Mertzios, Paul G. Spirakis, and Viktor Zamaraev. Temporal vertex cover with a sliding time window. Journal of Computer and System Sciences, 107:108-123, 2020. URL: https://doi.org/10.1016/j.jcss.2019.08.002.
  7. Steve Alpern. The rendezvous search problem. SIAM Journal on Control and Optimization, 33(3):673-683, 1995. URL: https://doi.org/10.1137/S0363012993249195.
  8. Steve Alpern. Rendezvous search: A personal perspective. Operations Research, 50(5):772-795, 2002. URL: https://doi.org/10.1287/opre.50.5.772.363.
  9. Steve Alpern and Shmuel Gal. The theory of search games and rendezvous, volume 55 of International Series in Operations Research & Management Science. Springer Science & Business Media, 2006. URL: https://doi.org/10.1007/b100809.
  10. Steven Alpern. Hide and seek games. In Seminar, Institut für höhere Studien, Wien, volume 26, 1976. Google Scholar
  11. K. Balasubramanian. Symmetry groups of chemical graphs. International Journal of Quantum Chemistry, 21(2):411-418, 1982. URL: https://doi.org/10.1002/qua.560210206.
  12. Fabian Ball and Andreas Geyer-Schulz. How symmetric are real-world graphs? A large-scale study. Symmetry, 10(1), 2018. URL: https://doi.org/10.3390/sym10010029.
  13. Subhash Bhagat and Andrzej Pelc. Deterministic rendezvous in infinite trees. CoRR, abs/2203.05160, 2022. URL: https://doi.org/10.48550/arXiv.2203.05160.
  14. Hans L. Bodlaender and Tom C. van der Zanden. On exploring always-connected temporal graphs of small pathwidth. Information Processing Letters, 142:68-71, 2019. URL: https://doi.org/10.1016/j.ipl.2018.10.016.
  15. John Adrian Bondy and Uppaluri Siva Ramachandra Murty. Graph theory with applications, volume 290. Macmillan London, 1976. Google Scholar
  16. Marjorie Bournat, Swan Dubois, and Franck Petit. Gracefully degrading gathering in dynamic rings. In Stabilization, Safety, and Security of Distributed Systems: 20th International Symposium, SSS 2018, Tokyo, Japan, November 4-7, 2018, Proceedings 20, pages 349-364. Springer, 2018. URL: https://doi.org/10.1007/978-3-030-03232-6_23.
  17. Benjamin Merlin Bumpus and Kitty Meeks. Edge exploration of temporal graphs. Algorithmica, 85(3):688-716, 2023. URL: https://doi.org/10.1007/s00453-022-01018-7.
  18. Arnaud Casteigts. Efficient generation of simple temporal graphs up to isomorphism. Github repository, 2020. URL: https://github.com/acasteigts/STGen.
  19. Arnaud Casteigts, Paola Flocchini, Walter Quattrociocchi, and Nicola Santoro. Time-varying graphs and dynamic networks. Int. J. Parallel Emergent Distributed Syst., 27(5):387-408, 2012. URL: https://doi.org/10.1080/17445760.2012.668546.
  20. Jurek Czyzowicz, Adrian Kosowski, and Andrzej Pelc. How to meet when you forget: Log-space rendezvous in arbitrary graphs. In Proceedings of the 29th ACM SIGACT-SIGOPS Symposium on Principles of Distributed Computing, PODC '10, pages 450-459, New York, NY, USA, 2010. Association for Computing Machinery. URL: https://doi.org/10.1145/1835698.1835801.
  21. Shantanu Das, Giuseppe Di Luna, Linda Pagli, and Giuseppe Prencipe. Compacting and grouping mobile agents on dynamic rings. In International Conference on Theory and Applications of Models of Computation, pages 114-133. Springer, 2019. URL: https://doi.org/10.1007/978-3-030-14812-6_8.
  22. Giuseppe Antonio Di Luna. Mobile agents on dynamic graphs. Distributed Computing by Mobile Entities: Current Research in Moving and Computing, pages 549-584, 2019. URL: https://doi.org/10.1007/978-3-030-11072-7_20.
  23. Giuseppe Antonio Di Luna, Paola Flocchini, Linda Pagli, Giuseppe Prencipe, Nicola Santoro, and Giovanni Viglietta. Gathering in dynamic rings. Theoretical Computer Science, 811:79-98, 2020. URL: https://doi.org/10.1016/j.tcs.2018.10.018.
  24. Yoann Dieudonné, Andrzej Pelc, and Vincent Villain. How to meet asynchronously at polynomial cost. In Proceedings of the 2013 ACM Symposium on Principles of Distributed Computing, pages 92-99, 2013. URL: https://doi.org/10.1137/130931990.
  25. Konstantinos Dogeas, Thomas Erlebach, Frank Kammer, Johannes Meintrup, and William K. Moses Jr au2. Exploiting automorphisms of temporal graphs for fast exploration and rendezvous, 2023. URL: https://arxiv.org/abs/2312.07140.
  26. Jessica Enright, Kitty Meeks, George B. Mertzios, and Viktor Zamaraev. Deleting edges to restrict the size of an epidemic in temporal networks. Journal of Computer and System Sciences, 119:60-77, 2021. URL: https://doi.org/10.1016/j.jcss.2021.01.007.
  27. Thomas Erlebach, Michael Hoffmann, and Frank Kammer. On temporal graph exploration. J. Comput. Syst. Sci., 119:1-18, 2021. URL: https://doi.org/10.1016/j.jcss.2021.01.005.
  28. Thomas Erlebach, Frank Kammer, Kelin Luo, Andrej Sajenko, and Jakob T. Spooner. Two moves per time step make a difference. In 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019). Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2019. URL: https://doi.org/10.4230/LIPIcs.ICALP.2019.141.
  29. Thomas Erlebach and Jakob T. Spooner. A game of cops and robbers on graphs with periodic edge-connectivity. In 46th International Conference on Current Trends in Theory and Practice of Informatics (SOFSEM 2020), volume 12011 of Lecture Notes in Computer Science, pages 64-75. Springer, 2020. URL: https://doi.org/10.1007/978-3-030-38919-2_6.
  30. Thomas Erlebach and Jakob T. Spooner. Exploration of k-edge-deficient temporal graphs. Acta Informatica, 59(4):387-407, 2022. URL: https://doi.org/10.1007/s00236-022-00421-5.
  31. Thomas Erlebach and Jakob T. Spooner. Parameterized temporal exploration problems. In 1st Symposium on Algorithmic Foundations of Dynamic Networks, SAND 2022, March 28-30, 2022, Virtual Conference, volume 221 of LIPIcs, pages 15:1-15:17. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2022. URL: https://doi.org/10.4230/LIPIcs.SAND.2022.15.
  32. Paola Flocchini. Distributed Computing by Mobile Entities: Current Research in Moving and Computing. Springer, 2019. URL: https://doi.org/10.1007/978-3-030-11072-7.
  33. Till Fluschnik, Hendrik Molter, Rolf Niedermeier, Malte Renken, and Philipp Zschoche. As time goes by: Reflections on treewidth for temporal graphs. In Treewidth, Kernels, and Algorithms: Essays Dedicated to Hans L. Bodlaender on the Occasion of His 60th Birthday, pages 49-77. Springer International Publishing, 2020. URL: https://doi.org/10.1007/978-3-030-42071-0_6.
  34. Till Fluschnik, Hendrik Molter, Rolf Niedermeier, Malte Renken, and Philipp Zschoche. Temporal graph classes: A view through temporal separators. Theoretical Computer Science, 806:197-218, 2020. URL: https://doi.org/10.1016/j.tcs.2019.03.031.
  35. Chris Godsil and Gordon F. Royle. Algebraic Graph Theory. Number Book 207 in Graduate Texts in Mathematics. Springer, 2001. URL: https://doi.org/10.1007/978-1-4613-0163-9.
  36. Thekla Hamm, Nina Klobas, George B. Mertzios, and Paul G. Spirakis. The complexity of temporal vertex cover in small-degree graphs. Proceedings of the AAAI Conference on Artificial Intelligence, 36(9):10193-10201, June 2022. URL: https://doi.org/10.1609/aaai.v36i9.21259.
  37. David Ilcinkas, Ralf Klasing, and Ahmed Mouhamadou Wade. Exploration of constantly connected dynamic graphs based on cactuses. In International Colloquium on Structural Information and Communication Complexity, pages 250-262. Springer, 2014. URL: https://doi.org/10.1007/978-3-319-09620-9_20.
  38. David Ilcinkas and Ahmed Mouhamadou Wade. Exploration of the t-interval-connected dynamic graphs: the case of the ring. In International Colloquium on Structural Information and Communication Complexity, pages 13-23. Springer, 2013. URL: https://doi.org/10.1007/978-3-319-03578-9_2.
  39. Ulrich Knauer and Kolja Knauer. Algebraic graph theory: morphisms, monoids and matrices, volume 41. Walter de Gruyter GmbH & Co KG, 2019. URL: https://doi.org/10.1515/9783110617368.
  40. Evangelos Kranakis, Danny Krizanc, and Euripides Marcou. The mobile agent rendezvous problem in the ring. Springer Nature, 2022. URL: https://doi.org/10.1007/978-3-031-01999-9.
  41. Evangelos Kranakis, Danny Krizanc, and Sergio Rajsbaum. Mobile agent rendezvous: A survey. In Structural Information and Communication Complexity, pages 1-9. Springer Berlin Heidelberg, 2006. URL: https://doi.org/10.1007/11780823_1.
  42. Josef Lauri and Raffaele Scapellato. Topics in Graph Automorphisms and Reconstruction. Cambridge University Press, 2016. URL: https://doi.org/10.1017/CBO9781316669846.
  43. Gianluca De Marco, Luisa Gargano, Evangelos Kranakis, Danny Krizanc, Andrzej Pelc, and Ugo Vaccaro. Asynchronous deterministic rendezvous in graphs. Theoretical Computer Science, 355(3):315-326, 2006. URL: https://doi.org/10.1016/j.tcs.2005.12.016.
  44. Andrea Marino and Ana Silva. Königsberg sightseeing: Eulerian walks in temporal graphs. In Combinatorial Algorithms, pages 485-500. Springer International Publishing, 2021. URL: https://doi.org/10.1007/978-3-030-79987-8_34.
  45. Andrea Marino and Ana Silva. Coloring temporal graphs. Journal of Computer and System Sciences, 123:171-185, 2022. URL: https://doi.org/10.1016/j.jcss.2021.08.004.
  46. George B. Mertzios, Hendrik Molter, Rolf Niedermeier, Viktor Zamaraev, and Philipp Zschoche. Computing Maximum Matchings in Temporal Graphs. In 37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020), volume 154 of Leibniz International Proceedings in Informatics (LIPIcs), pages 27:1-27:14, Dagstuhl, Germany, 2020. Schloss Dagstuhl - Leibniz-Zentrum für Informatik. URL: https://doi.org/10.4230/LIPIcs.STACS.2020.27.
  47. George B. Mertzios, Hendrik Molter, and Viktor Zamaraev. Sliding window temporal graph coloring. Journal of Computer and System Sciences, 120:97-115, 2021. URL: https://doi.org/10.1016/j.jcss.2021.03.005.
  48. Othon Michail. An introduction to temporal graphs: An algorithmic perspective. Internet Math., 12(4):239-280, 2016. URL: https://doi.org/10.1080/15427951.2016.1177801.
  49. Othon Michail and Paul G. Spirakis. Traveling salesman problems in temporal graphs. Computer Science (MFCS), 2014. Google Scholar
  50. Othon Michail and Paul G. Spirakis. Traveling salesman problems in temporal graphs. Theoretical Computer Science, 634:1-23, 2016. URL: https://doi.org/10.1016/j.tcs.2016.04.006.
  51. Othon Michail, Paul G. Spirakis, and Michail Theofilatos. Beyond rings: Gathering in 1-interval connected graphs. Parallel Processing Letters, 31(04):2150020, 2021. URL: https://doi.org/10.1142/S0129626421500201.
  52. Nils Morawietz, Carolin Rehs, and Mathias Weller. A Timecop’s Work Is Harder Than You Think. In 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020), volume 170 of Leibniz International Proceedings in Informatics (LIPIcs), pages 71:1-71:14, Dagstuhl, Germany, 2020. Schloss Dagstuhl - Leibniz-Zentrum für Informatik. URL: https://doi.org/10.4230/LIPIcs.MFCS.2020.71.
  53. Fukuhito Ooshita and Ajoy K. Datta. Brief announcement: feasibility of weak gathering in connected-over-time dynamic rings. In Stabilization, Safety, and Security of Distributed Systems: 20th International Symposium, SSS 2018, Tokyo, Japan, November 4-7, 2018, Proceedings 20, pages 393-397. Springer, 2018. URL: https://doi.org/10.1007/978-3-030-03232-6_27.
  54. Andrzej Pelc. Deterministic rendezvous in networks: A comprehensive survey. Networks, 59(3):331-347, 2012. URL: https://doi.org/10.1002/net.21453.
  55. Andrzej Pelc. Deterministic rendezvous algorithms. In Distributed Computing by Mobile Entities: Current Research in Moving and Computing, pages 423-454. Springer, 2019. URL: https://doi.org/10.1007/978-3-030-11072-7_17.
  56. Andrzej Pelc. Deterministic rendezvous algorithms. CoRR, abs/2303.10391, 2023. URL: https://doi.org/10.48550/arXiv.2303.10391.
  57. Philipp Plamper, Oliver J. Lechtenfeld, Peter Herzsprung, and Anika Groß. A temporal graph model to predict chemical transformations in complex dissolved organic matter. Environmental Science & Technology, 2023. URL: https://doi.org/10.1021/acs.est.3c00351.
  58. Nicola Santoro. Time to change: On distributed computing in dynamic networks (keynote). In 19th International Conference on Principles of Distributed Systems, OPODIS 2015, December 14-17, 2015, Rennes, France, volume 46 of LIPIcs, pages 3:1-3:14. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2015. URL: https://doi.org/10.4230/LIPIcs.OPODIS.2015.3.
  59. C. Shannon. Presentation of a maze solving machine. In Trans. 8th Conf. Cybernetics: Circular, Causal and Feedback Mechanisms in Biological and Social Systems (New York, 1951), pages 169-181, 1951. Google Scholar
  60. Masahiro Shibata, Naoki Kitamura, Ryota Eguchi, Yuichi Sudo, Junya Nakamura, and Yonghwan Kim. Partial gathering of mobile agents in dynamic tori. In 2nd Symposium on Algorithmic Foundations of Dynamic Networks (SAND 2023). Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2023. URL: https://doi.org/10.4230/LIPIcs.SAND.2023.2.
  61. Masahiro Shibata, Yuichi Sudo, Junya Nakamura, and Yonghwan Kim. Partial gathering of mobile agents in dynamic rings. In Stabilization, Safety, and Security of Distributed Systems: 23rd International Symposium, SSS 2021, Virtual Event, November 17-20, 2021, Proceedings 23, pages 440-455. Springer, 2021. URL: https://doi.org/10.1007/978-3-030-91081-5_29.
  62. Shadi Taghian Alamouti. Exploring temporal cycles and grids. PhD thesis, Concordia University, 2020. Google Scholar
  63. Philipp Zschoche, Till Fluschnik, Hendrik Molter, and Rolf Niedermeier. The complexity of finding small separators in temporal graphs. Journal of Computer and System Sciences, 107:72-92, 2020. URL: https://doi.org/10.1016/j.jcss.2019.07.006.