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Temporal Graph Realization from Fastest Paths

Authors Nina Klobas , George B. Mertzios , Hendrik Molter , Paul G. Spirakis

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Author Details

Nina Klobas
  • Department of Computer Science, Durham University, UK
George B. Mertzios
  • Department of Computer Science, Durham University, UK
Hendrik Molter
  • Department of Computer Science, Ben-Gurion University of the Negev, Beer-Sheva, Israel
Paul G. Spirakis
  • Department of Computer Science, University of Liverpool, UK

Funding

  • Mertzios, George B.: Supported by the EPSRC grant EP/P020372/1.
  • Molter, Hendrik: Supported by the ISF, grant nr. 1456/18, and by the European Union’s Horizon Europe research and innovation programme under grant agreement 949707.
  • Spirakis, Paul G.: Supported by the EPSRC grant EP/P02002X/1.

Cite As Get BibTex

Nina Klobas, George B. Mertzios, Hendrik Molter, and Paul G. Spirakis. Temporal Graph Realization from Fastest Paths. In 3rd Symposium on Algorithmic Foundations of Dynamic Networks (SAND 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 292, pp. 16:1-16:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.SAND.2024.16

Abstract

In this paper we initiate the study of the temporal graph realization problem with respect to the fastest path durations among its vertices, while we focus on periodic temporal graphs. Given an n × n matrix D and a Δ ∈ ℕ, the goal is to construct a Δ-periodic temporal graph with n vertices such that the duration of a fastest path from v_i to v_j is equal to D_{i,j}, or to decide that such a temporal graph does not exist. The variations of the problem on static graphs has been well studied and understood since the 1960’s (e.g. [Erdős and Gallai, 1960], [Hakimi and Yau, 1965]).
As it turns out, the periodic temporal graph realization problem has a very different computational complexity behavior than its static (i. e., non-temporal) counterpart. First we show that the problem is NP-hard in general, but polynomial-time solvable if the so-called underlying graph is a tree. Building upon those results, we investigate its parameterized computational complexity with respect to structural parameters of the underlying static graph which measure the "tree-likeness". We prove a tight classification between such parameters that allow fixed-parameter tractability (FPT) and those which imply W[1]-hardness. We show that our problem is W[1]-hard when parameterized by the feedback vertex number (and therefore also any smaller parameter such as treewidth, degeneracy, and cliquewidth) of the underlying graph, while we show that it is in FPT when parameterized by the feedback edge number (and therefore also any larger parameter such as maximum leaf number) of the underlying graph.

Subject Classification

ACM Subject Classification
  • Theory of computation → Graph algorithms analysis
  • Mathematics of computing → Discrete mathematics
Keywords
  • Temporal graph
  • periodic temporal labeling
  • fastest temporal path
  • graph realization
  • temporal connectivity
  • parameterized complexity

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References

  1. Eleni C Akrida, Leszek Gąsieniec, George B. Mertzios, and Paul G Spirakis. The complexity of optimal design of temporally connected graphs. Theory of Computing Systems, 61:907-944, 2017. Google Scholar
  2. 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. Google Scholar
  3. Emmanuel Arrighi, Niels Grüttemeier, Nils Morawietz, Frank Sommer, and Petra Wolf. Multi-parameter analysis of finding minors and subgraphs in edge-periodic temporal graphs. In Proceedings of the 48th International Conference on Current Trends in Theory and Practice of Computer Science (SOFSEM), pages 283-297, 2023. Google Scholar
  4. John Augustine, Keerti Choudhary, Avi Cohen, David Peleg, Sumathi Sivasubramaniam, and Suman Sourav. Distributed graph realizations. IEEE Transactions on Parallel and Distributed Systems, 33(6):1321-1337, 2022. Google Scholar
  5. Amotz Bar-Noy, Keerti Choudhary, David Peleg, and Dror Rawitz. Efficiently realizing interval sequences. SIAM Journal on Discrete Mathematics, 34(4):2318-2337, 2020. Google Scholar
  6. Amotz Bar-Noy, Keerti Choudhary, David Peleg, and Dror Rawitz. Graph realizations: Maximum degree in vertex neighborhoods. In Proceedings of the 17th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT), pages 10:1-10:17, 2020. Google Scholar
  7. Amotz Bar-Noy, David Peleg, Mor Perry, and Dror Rawitz. Composed degree-distance realizations of graphs. In Proceedings of the 32nd International Workshop on Combinatorial Algorithms (IWOCA), pages 63-77, 2021. Google Scholar
  8. Amotz Bar-Noy, David Peleg, Mor Perry, and Dror Rawitz. Graph realization of distance sets. In Proceedings of the 47th International Symposium on Mathematical Foundations of Computer Science (MFCS), pages 13:1-13:14, 2022. Google Scholar
  9. Mehdi Behzad and James E Simpson. Eccentric sequences and eccentric sets in graphs. Discrete Mathematics, 16(3):187-193, 1976. Google Scholar
  10. Robert E Bixby and Donald K Wagner. An almost linear-time algorithm for graph realization. Mathematics of Operations Research, 13(1):99-123, 1988. Google Scholar
  11. Binh-Minh Bui-Xuan, Afonso Ferreira, and Aubin Jarry. Computing shortest, fastest, and foremost journeys in dynamic networks. International Journal of Foundations of Computer Science, 14(02):267-285, 2003. Google Scholar
  12. Arnaud Casteigts, Timothée Corsini, and Writika Sarkar. Invited paper: Simple, strict, proper, happy: A study of reachability in temporal graphs. In Proceedings of the 24th International Symposium on Stabilization, Safety, and Security of Distributed Systems (SSS), pages 3-18, 2022. Google Scholar
  13. Arnaud Casteigts, Paola Flocchini, Walter Quattrociocchi, and Nicola Santoro. Time-varying graphs and dynamic networks. International Journal of Parallel, Emergent and Distributed Systems, 27(5):387-408, 2012. Google Scholar
  14. Arnaud Casteigts, Anne-Sophie Himmel, Hendrik Molter, and Philipp Zschoche. Finding temporal paths under waiting time constraints. Algorithmica, 83(9):2754-2802, 2021. Google Scholar
  15. Wai-Kai Chen. On the realization of a (p, s)-digraph with prescribed degrees. Journal of the Franklin Institute, 281(5):406-422, 1966. Google Scholar
  16. Fan Chung, Mark Garrett, Ronald Graham, and David Shallcross. Distance realization problems with applications to internet tomography. Journal of Computer and System Sciences, 63(3):432-448, 2001. Google Scholar
  17. Joseph C. Culberson and Piotr Rudnicki. A fast algorithm for constructing trees from distance matrices. Information Processing Letters, 30(4):215-220, 1989. Google Scholar
  18. Argyrios Deligkas and Igor Potapov. Optimizing reachability sets in temporal graphs by delaying. Information and Computation, 285:104890, 2022. Google Scholar
  19. 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. Google Scholar
  20. Jessica Enright, Kitty Meeks, and Fiona Skerman. Assigning times to minimise reachability in temporal graphs. Journal of Computer and System Sciences, 115:169-186, 2021. Google Scholar
  21. Jessica A. Enright, Kitty Meeks, and Hendrik Molter. Counting temporal paths. In Proceedings of the 40th International Symposium on Theoretical Aspects of Computer Science (STACS), volume 254, pages 30:1-30:19, 2023. Google Scholar
  22. Paul Erdős and Tibor Gallai. Graphs with prescribed degrees of vertices. Mat. Lapok, 11:264-274, 1960. Google Scholar
  23. Thomas Erlebach, Michael Hoffmann, and Frank Kammer. On temporal graph exploration. Journal of Computer and System Sciences, 119:1-18, 2021. Google Scholar
  24. Thomas Erlebach, Nils Morawietz, and Petra Wolf. Parameterized algorithms for multi-label periodic temporal graph realization. In Proceedings of the 3rd Symposium on Algorithmic Foundations of Dynamic Networks (SAND), pages 12:1-12:16, 2024. URL: https://doi.org/10.4230/LIPIcs.SAND.2024.12.
  25. Thomas Erlebach and Jakob T. Spooner. A game of cops and robbers on graphs with periodic edge-connectivity. In Proceedings of the 46th International Conference on Current Trends in Theory and Practice of Informatics (SOFSEM), pages 64-75, 2020. Google Scholar
  26. Michael R. Fellows, Danny Hermelin, Frances Rosamond, and Stéphane Vialette. On the parameterized complexity of multiple-interval graph problems. Theoretical Computer Science, 410(1):53-61, 2009. Google Scholar
  27. Michael R. Fellows, Bart M. P. Jansen, and Frances A. Rosamond. Towards fully multivariate algorithmics: Parameter ecology and the deconstruction of computational complexity. European Journal of Combinatorics, 34(3):541-566, 2013. Google Scholar
  28. 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. Google Scholar
  29. András Frank. Augmenting graphs to meet edge-connectivity requirements. SIAM Journal on Discrete Mathematics, 5(1):25-53, 1992. Google Scholar
  30. András Frank. Connectivity augmentation problems in network design. Mathematical Programming: State of the Art 1994, 1994. Google Scholar
  31. H. Frank and Wushow Chou. Connectivity considerations in the design of survivable networks. IEEE Transactions on Circuit Theory, 17(4):486-490, 1970. Google Scholar
  32. Eugen Füchsle, Hendrik Molter, Rolf Niedermeier, and Malte Renken. Delay-robust routes in temporal graphs. In Proceedings of the 39th International Symposium on Theoretical Aspects of Computer Science (STACS), pages 30:1-30:15, 2022. Google Scholar
  33. Eugen Füchsle, Hendrik Molter, Rolf Niedermeier, and Malte Renken. Temporal connectivity: Coping with foreseen and unforeseen delays. In Proceedings of the 1st Symposium on Algorithmic Foundations of Dynamic Networks (SAND), pages 17:1-17:17, 2022. Google Scholar
  34. D.R. Fulkerson. Zero-one matrices with zero trace. Pacific Journal of Mathematics, 10(3):831-836, 1960. Google Scholar
  35. Petr A. Golovach and George B. Mertzios. Graph editing to a given degree sequence. Theoretical Computer Science, 665:1-12, 2017. Google Scholar
  36. Martin Charles Golumbic and Ann N. Trenk. Tolerance Graphs. Cambridge Studies in Advanced Mathematics. Cambridge University Press, 2004. Google Scholar
  37. Ralph E Gomory and Tien Chung Hu. Multi-terminal network flows. Journal of the Society for Industrial and Applied Mathematics, 9(4):551-570, 1961. Google Scholar
  38. Martin Grötschel, Clyde L Monma, and Mechthild Stoer. Design of survivable networks. Handbooks in Operations Research and Management Science, 7:617-672, 1995. Google Scholar
  39. Jiong Guo, Falk Hüffner, and Rolf Niedermeier. A structural view on parameterizing problems: Distance from triviality. In Proceedings of the 1st International Workshop on Parameterized and Exact Computation (IWPEC), pages 162-173, 2004. Google Scholar
  40. S. Louis Hakimi. On realizability of a set of integers as degrees of the vertices of a linear graph. I. Journal of the Society for Industrial and Applied Mathematics, 10(3):496-506, 1962. Google Scholar
  41. S. Louis Hakimi and Stephen S. Yau. Distance matrix of a graph and its realizability. Quarterly of applied mathematics, 22(4):305-317, 1965. Google Scholar
  42. Pavol Hell and David Kirkpatrick. Linear-time certifying algorithms for near-graphical sequences. Discrete Mathematics, 309(18):5703-5713, 2009. Google Scholar
  43. David Kempe, Jon M. Kleinberg, and Amit Kumar. Connectivity and inference problems for temporal networks. Journal of Computer and System Sciences, 64(4):820-842, 2002. Google Scholar
  44. Nina Klobas, George B. Mertzios, Hendrik Molter, Rolf Niedermeier, and Philipp Zschoche. Interference-free walks in time: Temporally disjoint paths. Autonomous Agents and Multi-Agent Systems, 37(1):1, 2023. Google Scholar
  45. Nina Klobas, George B. Mertzios, Hendrik Molter, and Paul G. Spirakis. The complexity of computing optimum labelings for temporal connectivity. In Proceedings of the 47th International Symposium on Mathematical Foundations of Computer Science (MFCS), pages 62:1-62:15, 2022. Google Scholar
  46. Nina Klobas, George B. Mertzios, Hendrik Molter, and Paul G. Spirakis. Realizing temporal graphs from fastest travel times. CoRR, abs/2302.08860, 2023. URL: https://doi.org/10.48550/arXiv.2302.08860.
  47. Fabian Kuhn and Rotem Oshman. Dynamic networks: Models and algorithms. SIGACT News, 42(1):82-96, March 2011. Google Scholar
  48. Pascal Kunz, Hendrik Molter, and Meirav Zehavi. In which graph structures can we efficiently find temporally disjoint paths and walks? In Proceedings of the 32nd International Joint Conference on Artificial Intelligence (IJCAI), pages 180-188, 2023. Google Scholar
  49. Hendrik W. Lenstra. Integer programming with a fixed number of variables. Mathematics of Operations Research, 8:538-548, 1983. Google Scholar
  50. Jure Leskovec and Andrej Krevl. SNAP Datasets: Stanford large network dataset collection. http://snap.stanford.edu/data, June 2014.
  51. Linda Lesniak. Eccentric sequences in graphs. Periodica Mathematica Hungarica, 6:287-293, 1975. Google Scholar
  52. Ross M. McConnell and Jeremy P. Spinrad. Construction of probe interval models. In Proceedings of the 13th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 866-875, 2002. Google Scholar
  53. F.R. McMorris, Chi Wang, and Peisen Zhang. On probe interval graphs. Discrete Applied Mathematics, 88(1):315-324, 1998. Computational Molecular Biology DAM - CMB Series. Google Scholar
  54. George B. Mertzios, Othon Michail, and Paul G. Spirakis. Temporal network optimization subject to connectivity constraints. Algorithmica, 81(4):1416-1449, 2019. Google Scholar
  55. George B. Mertzios, Hendrik Molter, Malte Renken, Paul G. Spirakis, and Philipp Zschoche. The complexity of transitively orienting temporal graphs. In Proceedings of the 46th International Symposium on Mathematical Foundations of Computer Science (MFCS), pages 75:1-75:18, 2021. Google Scholar
  56. George B. Mertzios, Hendrik Molter, and Paul G. Spirakis. Realizing temporal transportation trees. CoRR, abs/2403.18513, 2024. URL: https://doi.org/10.48550/arXiv.2403.18513.
  57. Hendrik Molter, Malte Renken, and Philipp Zschoche. Temporal reachability minimization: Delaying vs. deleting. In Proceedings of the 46th International Symposium on Mathematical Foundations of Computer Science (MFCS), pages 76:1-76:15, 2021. Google Scholar
  58. Nils Morawietz, Carolin Rehs, and Mathias Weller. A timecop’s work is harder than you think. In Proceedings of the 45th International Symposium on Mathematical Foundations of Computer Science (MFCS), volume 170, pages 71-1, 2020. Google Scholar
  59. Nils Morawietz and Petra Wolf. A timecop’s chase around the table. In Proceedings of the 46th International Symposium on Mathematical Foundations of Computer Science (MFCS), 2021. Google Scholar
  60. A.N. Patrinos and S. Louis Hakimi. The distance matrix of a graph and its tree realization. Quarterly of Applied Mathematics, 30:255-269, 1972. Google Scholar
  61. Elena Rubei. Weighted graphs with distances in given ranges. Journal of Classification, 33:282-297, 2016. Google Scholar
  62. Piotr Sapiezynski, Arkadiusz Stopczynski, Radu Gatej, and Sune Lehmann. Tracking human mobility using wifi signals. PloS one, 10(7):e0130824, 2015. Google Scholar
  63. H. Tamura, M. Sengoku, S. Shinoda, and T. Abe. Realization of a network from the upper and lower bounds of the distances (or capacities) between vertices. In Proceedings of the 1993 IEEE International Symposium on Circuits and Systems (ISCAS), pages 2545-2548, 1993. Google Scholar
  64. Huanhuan Wu, James Cheng, Yiping Ke, Silu Huang, Yuzhen Huang, and Hejun Wu. Efficient algorithms for temporal path computation. IEEE Transactions on Knowledge and Data Engineering, 28(11):2927-2942, 2016. Google Scholar
  65. Philipp Zschoche, Till Fluschnik, Hendrik Molter, and Rolf Niedermeier. The complexity of finding separators in temporal graphs. Journal of Computer and System Sciences, 107:72-92, 2020. Google Scholar
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