LIPIcs.SAND.2024.12.pdf
- Filesize: 0.77 MB
- 16 pages
Document
Parameterized Algorithms for Multi-Label Periodic Temporal Graph Realization
Authors
Thomas Erlebach 
,
Nils Morawietz ,
Petra Wolf
-
Part of:
Volume:
3rd Symposium on Algorithmic Foundations of Dynamic Networks (SAND 2024)
Series: Leibniz International Proceedings in Informatics (LIPIcs)
Conference: Symposium on Algorithmic Foundations of Dynamic Networks (SAND) - License:
Creative Commons Attribution 4.0 International license
- Publication Date: 2024-05-31
File
Document Identifiers
Author Details
Cite As Get BibTex
Thomas Erlebach, Nils Morawietz, and Petra Wolf. Parameterized Algorithms for Multi-Label Periodic Temporal Graph Realization. In 3rd Symposium on Algorithmic Foundations of Dynamic Networks (SAND 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 292, pp. 12:1-12:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.SAND.2024.12
Abstract
In the periodic temporal graph realization problem introduced by Klobas et al. [SAND '24] one is given a period Δ and an n× n matrix D of desired fastest travel times, and the task is to decide if there is a simple periodic temporal graph with period Δ such that the fastest travel time between any pair of vertices matches the one specified by D. We generalize the problem from simple temporal graphs to temporal graphs where each edge can appear up to 𝓁 times in each period, for some given integer 𝓁. For the resulting problem Multi-Label Periodic TGR, we show that it is fixed-parameter tractable for parameter n and for parameter vc+Δ, where vc is the vertex cover number of the underlying graph. We also show the existence of a polynomial kernel for parameter nu+d_max, where nu is the number of non-universal vertices of the underlying graph and d_max is the largest entry of D. Furthermore, we show that the problem is NP-hard for each 𝓁 ≥ 5, even if the underlying graph is a tree, a case that was known to be solvable in polynomial time if the task is to construct a simple periodic temporal graph, that is, if 𝓁 = 1.
Subject Classification
ACM Subject Classification
- Theory of computation → Design and analysis of algorithms
Keywords
- Fixed-Parameter Tractability
- Almost-Clique
- Kernelization
- Dynamic Network
- Temporal Graph
Metrics
- Access Statistics
-
Total Accesses (updated on a weekly basis)
0PDF Downloads
References
- Eleni C. Akrida, Leszek Gasieniec, George B. Mertzios, and Paul G. Spirakis. The complexity of optimal design of temporally connected graphs. Theory Comput. Syst., 61(3):907-944, 2017. URL: https://doi.org/10.1007/S00224-017-9757-X.
- Emmanuel Arrighi, Niels Grüttemeier, Nils Morawietz, Frank Sommer, and Petra Wolf. Multi-parameter analysis of finding minors and subgraphs in edge periodic temporal graphs. CoRR, abs/2203.07401, 2022. URL: https://doi.org/10.48550/ARXIV.2203.07401.
- Binh-Minh Bui-Xuan, Afonso Ferreira, and Aubin Jarry. Computing shortest, fastest, and foremost journeys in dynamic networks. Int. J. Found. Comput. Sci., 14(2):267-285, 2003. URL: https://doi.org/10.1142/S0129054103001728.
- 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.
- Marek Cygan, Fedor V. Fomin, Lukasz Kowalik, Daniel Lokshtanov, Dániel Marx, Marcin Pilipczuk, Michal Pilipczuk, and Saket Saurabh. Parameterized Algorithms. Springer, 2015. URL: https://doi.org/10.1007/978-3-319-21275-3.
-
Reinhard Diestel. Graph Theory. Springer-Verlag, 2000.
- Rodney G. Downey and Michael R. Fellows. Fundamentals of Parameterized Complexity. Texts in Computer Science. Springer, 2013. URL: https://doi.org/10.1007/978-1-4471-5559-1.
- Jessica A. Enright, Kitty Meeks, and Fiona Skerman. Assigning times to minimise reachability in temporal graphs. J. Comput. Syst. Sci., 115:169-186, 2021. URL: https://doi.org/10.1016/J.JCSS.2020.08.001.
-
Paul Erdős and Tibor Gallai. Graphs with points of prescribed degree (in Hungarian). Mat. Lapok, 11:264-274, 1961.
-
S. L. Hakimi and S. S. Yau. Distance matrix of a graph and its realizability. Quarterly of Applied Mathematics, 22(4):305-317, 1965.
- Nina Klobas, George B. Mertzios, Hendrik Molter, and Paul G. Spirakis. The complexity of computing optimum labelings for temporal connectivity. In 47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022), volume 241 of LIPIcs, pages 62:1-62:15. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2022. URL: https://doi.org/10.4230/LIPICS.MFCS.2022.62.
- 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), volume 292 of LIPIcs, pages 16:1-16:18. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2024. URL: https://doi.org/10.4230/LIPIcs.SAND.2024.16.
- George B. Mertzios, Othon Michail, and Paul G. Spirakis. Temporal network optimization subject to connectivity constraints. Algorithmica, 81(4):1416-1449, 2019. URL: https://doi.org/10.1007/S00453-018-0478-6.
- Othon Michail and Paul G. Spirakis. Elements of the theory of dynamic networks. Commun. ACM, 61(2):72, 2018. URL: https://doi.org/10.1145/3156693.
- Owen J. Murphy. Computing independent sets in graphs with large girth. Discret. Appl. Math., 35(2):167-170, 1992. URL: https://doi.org/10.1016/0166-218X(92)90041-8.
- Victor Reis and Thomas Rothvoss. The subspace flatness conjecture and faster integer programming. In 64th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2023, Santa Cruz, CA, USA, November 6-9, 2023, pages 974-988. IEEE, 2023. URL: https://doi.org/10.1109/FOCS57990.2023.00060.
- Huanhuan Wu, James Cheng, Silu Huang, Yiping Ke, Yi Lu, and Yanyan Xu. Path problems in temporal graphs. Proc. VLDB Endow., 7(9):721-732, 2014. URL: https://doi.org/10.14778/2732939.2732945.