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Spanner Enumeration for Temporal Graphs

Authors Kazuhiro Kurita , Andrea Marino , Jason Schoeters , Takeaki Uno

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Subject Classification

ACM Subject Classification
  • Mathematics of computing → Graph enumeration
  • Theory of computation → Sparsification and spanners
  • Mathematics of computing → Graph algorithms
  • Theory of computation → Problems, reductions and completeness
  • Networks → Network dynamics
Keywords
  • temporal graphs
  • temporal spanners
  • one-to-all connectivity
  • all-to-all connectivity enumeration
  • NP-completeness
  • Dual-hardness
  • binary partition tree
  • flashlight search
  • polynomial delay

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Abstract

A spanner of a temporal graph is a subset of edges that preserves connectivity over time between vertices. A minimal spanner is one in which no additional edges can be removed without breaking this connectivity. Our focus is on enumerating minimal spanners for a given temporal graph. We explore several variations of this problem based on the type of connectivity that must be maintained, ranging from one-to-all connectivity to one-to-all-to-one, many-to-all, and finally all-to-all connectivity. We establish that these problems become progressively harder: (i) We present a polynomial-delay enumeration algorithm for one-to-all connectivity; (ii) We prove Dual-hardness for both one-to-all-to-one and many-to-all connectivity, even in the restricted case of two-to-all; (iii) Finally, for all-to-all connectivity, we show that enumeration cannot be performed in output-polynomial time unless P = NP.

Cite As Get BibTex

Kazuhiro Kurita, Andrea Marino, Jason Schoeters, and Takeaki Uno. Spanner Enumeration for Temporal Graphs. In 4th Symposium on Algorithmic Foundations of Dynamic Networks (SAND 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 330, pp. 9:1-9:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.SAND.2025.9

Author Details

Kazuhiro Kurita
  • Nagoya University, Nagoya, Japan
Andrea Marino
  • Dipartimento di Statistica, Informatica, Applicazioni, Università degli Studi di Firenze, Italy
Jason Schoeters
  • Dipartimento di Statistica, Informatica, Applicazioni, Università degli Studi di Firenze, Italy
Takeaki Uno
  • National Institute of Informatics, Tokyo, Japan

Funding

  • Kurita, Kazuhiro: JSPS Kakenhi Grant Numbers JP21K17812, JP23K24806, and JP25K21273, and JST ACT-X Grant Number JPMJAX2105.
  • Marino, Andrea: Italian PNRR CN4 Centro Nazionale per la Mobilità Sostenibile, NextGeneration EU - CUP, B13C22001000001. MUR of Italy, under PRIN Project n. 2022ME9Z78 - NextGRAAL: Next-generation algorithms for constrained GRAph visuALization, PRIN PNRR Project n. P2022NZPJA - DLT-FRUIT: A user centered framework for facilitating DLTs FRUITion.
  • Schoeters, Jason: PRIN PNRR Project n. P2022NZPJA - DLT-FRUIT: A user centered framework for facilitating DLTs FRUITion.

Acknowledgements

We would like to thank Lapo Cioni for the joining in the verification of constructions and proofs during coffee breaks.

References

  1. Reyan Ahmed, Greg Bodwin, Faryad Darabi Sahneh, Keaton Hamm, Mohammad Javad Latifi Jebelli, Stephen Kobourov, and Richard Spence. Graph spanners: A tutorial review. Computer Science Review, 37:100253, 2020. URL: https://doi.org/10.1016/J.COSREV.2020.100253.
  2. 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(3), 2017. URL: https://doi.org/10.1007/S00224-017-9757-X.
  3. Kyriakos Axiotis and Dimitris Fotakis. On the size and the approximability of minimum temporally connected subgraphs. In 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016). Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2016. Google Scholar
  4. Claude Berge. Hypergraphs: combinatorics of finite sets, volume 45. Elsevier, 1984. Google Scholar
  5. Davide Bilò, Gianlorenzo D'Angelo, Luciano Gualà, Stefano Leucci, and Mirko Rossi. Sparse temporal spanners with low stretch. In 30th Annual European Symposium on Algorithms (ESA 2022). Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2022. Google Scholar
  6. Davide Bilò, Gianlorenzo D'Angelo, Luciano Gualà, Stefano Leucci, and Mirko Rossi. Blackout-tolerant temporal spanners. Journal of Computer and System Sciences, 141:103495, 2024. URL: https://doi.org/10.1016/J.JCSS.2023.103495.
  7. E. Boros, K. Elbassioni, V. Gurvich, and L. Khachiyan. Enumerating minimal dicuts and strongly connected subgraphs and related geometric problems. In Daniel Bienstock and George Nemhauser, editors, Integer Programming and Combinatorial Optimization, pages 152-162, Berlin, Heidelberg, 2004. Springer Berlin Heidelberg. Google Scholar
  8. Endre Boros, Konrad Borys, Khaled Elbassioni, Vladimir Gurvich, Kazuhisa Makino, and Gabor Rudolf. Generating minimal k-vertex connected spanning subgraphs. In Guohui Lin, editor, Computing and Combinatorics, pages 222-231, Berlin, Heidelberg, 2007. Springer Berlin Heidelberg. Google Scholar
  9. Endre Boros and Kazuhisa Makino. Generating minimal redundant and maximal irredundant subhypergraphs. Discret. Appl. Math., 358:217-229, 2024. URL: https://doi.org/10.1016/J.DAM.2024.07.006.
  10. Caroline Brosse, Oscar Defrain, Kazuhiro Kurita, Vincent Limouzy, Takeaki Uno, and Kunihiro Wasa. On the hardness of inclusion-wise minimal separators enumeration. Inf. Process. Lett., 185:106469, 2024. URL: https://doi.org/10.1016/J.IPL.2023.106469.
  11. Daniela Bubboloni, Costanza Catalano, Andrea Marino, and Ana Silva. On computing optimal temporal branchings and spanning subgraphs. J. Comput. Syst. Sci., 148:103596, 2025. URL: https://doi.org/10.1016/J.JCSS.2024.103596.
  12. Fritz Bökler, Matthias Ehrgott, Christopher Morris, and Petra Mutzel. Output-sensitive complexity of multiobjective combinatorial optimization. Journal of Multi-Criteria Decision Analysis, 24(1-2):25-36, 2017. URL: https://doi.org/10.1002/mcda.1603.
  13. Arnaud Casteigts. Finding structure in dynamic networks. CoRR, abs/1807.07801, 75p, 2018. URL: http://arxiv.org/abs/1807.07801.
  14. Arnaud Casteigts and Timothée Corsini. In search of the lost tree: Hardness and relaxation of spanning trees in temporal graphs. In International Colloquium on Structural Information and Communication Complexity, pages 138-155. Springer, 2024. URL: https://doi.org/10.1007/978-3-031-60603-8_8.
  15. 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. URL: https://doi.org/10.1080/17445760.2012.668546.
  16. Arnaud Casteigts, Joseph G Peters, and Jason Schoeters. Temporal cliques admit sparse spanners. Journal of Computer and System Sciences, 121:1-17, 2021. URL: https://doi.org/10.1016/J.JCSS.2021.04.004.
  17. Arnaud Casteigts, Michael Raskin, Malte Renken, and Viktor Zamaraev. Sharp thresholds in random simple temporal graphs. SIAM Journal on Computing, 53(2):346-388, 2024. URL: https://doi.org/10.1137/22M1511916.
  18. Esteban Christiann, Eric Sanlaville, and Jason Schoeters. On inefficiently connecting temporal networks. In 3rd Symposium on Algorithmic Foundations of Dynamic Networks (SAND 2024). Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2024. Google Scholar
  19. Alessio Conte, Roberto Grossi, Mamadou Moustapha Kanté, Andrea Marino, Takeaki Uno, and Kunihiro Wasa. Listing induced steiner subgraphs as a compact way to discover steiner trees in graphs. In Peter Rossmanith, Pinar Heggernes, and Joost-Pieter Katoen, editors, 44th International Symposium on Mathematical Foundations of Computer Science, MFCS 2019, August 26-30, 2019, Aachen, Germany, volume 138 of LIPIcs, pages 73:1-73:14. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2019. URL: https://doi.org/10.4230/LIPICS.MFCS.2019.73.
  20. Edsger W Dijkstra. A note on two problems in connexion with graphs. In Edsger Wybe Dijkstra: his life, work, and legacy, pages 287-290. 2022. URL: https://doi.org/10.1145/3544585.3544600.
  21. Thomas Eiter, Kazuhisa Makino, and Georg Gottlob. Computational aspects of monotone dualization: A brief survey. Discrete Applied Mathematics, 156(11):2035-2049, 2008. In Memory of Leonid Khachiyan (1952 - 2005 ). URL: https://doi.org/10.1016/j.dam.2007.04.017.
  22. Khaled Elbassioni, Imran Rauf, and Saurabh Ray. A global parallel algorithm for enumerating minimal transversals of geometric hypergraphs. Theoretical Computer Science, 767:26-33, 2019. URL: https://doi.org/10.1016/J.TCS.2018.09.027.
  23. Hovhannes Harutyunyan, Arthur Liestman, Joseph Peters, and Dana Richards. Broadcasting and Gossiping. In Ping Zhang, editor, Handbook of Graph Theory, Second Edition, volume 20134658, pages 1477-1494. Chapman and Hall/CRC, December 2013. Series Title: Discrete Mathematics and Its Applications. URL: https://doi.org/10.1201/b16132-87.
  24. David Kempe, Jon Kleinberg, and Amit Kumar. Connectivity and inference problems for temporal networks. In Proceedings of the thirty-second annual ACM symposium on Theory of computing, pages 504-513, 2000. URL: https://doi.org/10.1145/335305.335364.
  25. Leonid Khachiyan, Endre Boros, Khaled Elbassioni, Vladimir Gurvich, and Kazuhisa Makino. Enumerating disjunctions and conjunctions of paths and cuts in reliability theory. Discrete Applied Mathematics, 155(2):137-149, 2007. 29th Symposium on Mathematical Foundations of Computer Science MFCS 2004. URL: https://doi.org/10.1016/j.dam.2006.04.032.
  26. Leonid Khachiyan, Endre Boros, Khaled M. Elbassioni, and Vladimir Gurvich. On enumerating minimal dicuts and strongly connected subgraphs. Algorithmica, 50(1):159-172, 2008. URL: https://doi.org/10.1007/S00453-007-9074-X.
  27. Benny Kimelfeld and Yehoshua Sagiv. Efficiently enumerating results of keyword search over data graphs. Inf. Syst., 33(4-5):335-359, 2008. URL: https://doi.org/10.1016/J.IS.2008.01.002.
  28. Yasuaki Kobayashi, Kazuhiro Kurita, Yasuko Matsui, and Hirotaka Ono. Enumerating minimal vertex covers and dominating sets with capacity and/or connectivity constraints. In Adele Anna Rescigno and Ugo Vaccaro, editors, Combinatorial Algorithms - 35th International Workshop, IWOCA 2024, Ischia, Italy, July 1-3, 2024, Proceedings, volume 14764 of Lecture Notes in Computer Science, pages 232-246. Springer, 2024. URL: https://doi.org/10.1007/978-3-031-63021-7_18.
  29. Yasuaki Kobayashi, Kazuhiro Kurita, and Kunihiro Wasa. Linear-delay enumeration for minimal steiner problems. In Leonid Libkin and Pablo Barceló, editors, PODS '22: International Conference on Management of Data, Philadelphia, PA, USA, June 12 - 17, 2022, pages 301-313. ACM, 2022. URL: https://doi.org/10.1145/3517804.3524148.
  30. Joseph B Kruskal. On the shortest spanning subtree of a graph and the traveling salesman problem. Proceedings of the American Mathematical society, 7(1):48-50, 1956. Google Scholar
  31. E. L. Lawler, J. K. Lenstra, and A. H. G. Rinnooy Kan. Generating all maximal independent sets: Np-hardness and polynomial-time algorithms. SIAM Journal on Computing, 9(3):558-565, 1980. URL: https://doi.org/10.1137/0209042.
  32. Arnaud Mary and Yann Strozecki. Efficient enumeration of solutions produced by closure operations. Discret. Math. Theor. Comput. Sci., 21(3), 2019. URL: https://doi.org/10.23638/DMTCS-21-3-22.
  33. Robert Clay Prim. Shortest connection networks and some generalizations. The Bell System Technical Journal, 36(6):1389-1401, 1957. Google Scholar
  34. Yann Strozecki. Enumeration complexity. Bull. EATCS, 129, 2019. URL: http://bulletin.eatcs.org/index.php/beatcs/article/view/596/605.
  35. B 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. URL: https://doi.org/10.1142/S0129054103001728.
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