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Partial Temporal Vertex Cover with Bounded Activity Intervals

Authors Riccardo Dondi , Fabrizio Montecchiani , Giacomo Ortali , Tommaso Piselli , Alessandra Tappini

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Author Details

Riccardo Dondi
  • Università degli Studi di Bergamo, Italy
Fabrizio Montecchiani
  • Università degli Studi di Perugia, Italy
Giacomo Ortali
  • Università degli Studi di Perugia, Italy
Tommaso Piselli
  • Università degli Studi di Perugia, Italy
Alessandra Tappini
  • Università degli Studi di Perugia, Italy

Funding

This work was partially supported by: (i) MUR PRIN Proj. 2022TS4Y3N - "EXPAND: scalable algorithms for EXPloratory Analyses of heterogeneous and dynamic Networked Data"; (ii) MUR PRIN Proj. 2022ME9Z78 - "NextGRAAL: Next-generation algorithms for constrained GRAph visuALization"; (iii) University of Perugia, Fondi di Ricerca di Ateneo, edizione 2021, project "AIDMIX - Artificial Intelligence for Decision making: Methods for Interpretability and eXplainability".

Cite AsGet BibTex

Riccardo Dondi, Fabrizio Montecchiani, Giacomo Ortali, Tommaso Piselli, and Alessandra Tappini. Partial Temporal Vertex Cover with Bounded Activity Intervals. In 3rd Symposium on Algorithmic Foundations of Dynamic Networks (SAND 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 292, pp. 11:1-11:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.SAND.2024.11

Abstract

Different variants of Vertex Cover have recently garnered attention in the context of temporal graphs. One of these variants is motivated by the need to summarize timeline activities in social networks. Here, the activities of individual vertices, representing users, are characterized by time intervals. In this paper, we explore a scenario where the temporal span of each vertex’s activity interval is bounded by an integer 𝓁, and the objective is to maximize the number of (temporal) edges that are covered. We establish the APX-hardness of this problem and the NP-hardness of the corresponding decision problem, even under the restricted condition where the temporal domain comprises only two timestamps and each edge appears at most once. Subsequently, we delve into the parameterized complexity of the problem, offering two fixed-parameter algorithms parameterized by: (i) the number k of temporal edges covered by the solution, and (ii) the number h of temporal edges not covered by the solution. Finally, we present a polynomial-time approximation algorithm achieving a factor of 3/4.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Graph algorithms
  • Theory of computation → Fixed parameter tractability
  • Theory of computation → Approximation algorithms analysis
Keywords
  • Temporal Graphs
  • Temporal Vertex Cover
  • Parameterized Complexity
  • Approximation Algorithms

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References

  1. Eleni C. Akrida, George B. Mertzios, Paul G. Spirakis, and Viktor Zamaraev. Temporal vertex cover with a sliding time window. J. Comput. Syst. Sci., 107:108-123, 2020. URL: https://doi.org/10.1016/j.jcss.2019.08.002.
  2. Piotr Berman and Marek Karpinski. On some tighter inapproximability results. Electron. Colloquium Comput. Complex., TR98-029, 1998. URL: https://arxiv.org/abs/TR98-029.
  3. Piotr Berman and Marek Karpinski. On some tighter inapproximability results (extended abstract). In Jirí Wiedermann, Peter van Emde Boas, and Mogens Nielsen, editors, Automata, Languages and Programming, 26th International Colloquium, ICALP'99, Prague, Czech Republic, July 11-15, 1999, Proceedings, volume 1644 of Lecture Notes in Computer Science, pages 200-209. Springer, 1999. URL: https://doi.org/10.1007/3-540-48523-6_17.
  4. Riccardo Dondi. Untangling temporal graphs of bounded degree. Theor. Comput. Sci., 969:114040, 2023. URL: https://doi.org/10.1016/J.TCS.2023.114040.
  5. Riccardo Dondi and Manuel Lafond. An FPT algorithm for temporal graph untangling. In Neeldhara Misra and Magnus Wahlström, editors, 18th International Symposium on Parameterized and Exact Computation, IPEC 2023, September 6-8, 2023, Amsterdam, The Netherlands, volume 285 of LIPIcs, pages 12:1-12:16. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2023. URL: https://doi.org/10.4230/LIPICS.IPEC.2023.12.
  6. Riccardo Dondi and Alexandru Popa. Timeline cover in temporal graphs: Exact and approximation algorithms. In Sun-Yuan Hsieh, Ling-Ju Hung, and Chia-Wei Lee, editors, Combinatorial Algorithms - 34th International Workshop, IWOCA 2023, Tainan, Taiwan, June 7-10, 2023, Proceedings, volume 13889 of Lecture Notes in Computer Science, pages 173-184. Springer, 2023. URL: https://doi.org/10.1007/978-3-031-34347-6_15.
  7. Vincent Froese, Pascal Kunz, and Philipp Zschoche. Disentangling the computational complexity of network untangling. In Luc De Raedt, editor, Proceedings of the Thirty-First International Joint Conference on Artificial Intelligence, IJCAI 2022, Vienna, Austria, 23-29 July 2022, pages 2037-2043. ijcai.org, 2022. URL: https://doi.org/10.24963/ijcai.2022/283.
  8. Rajiv Gandhi, Samir Khuller, and Aravind Srinivasan. Approximation algorithms for partial covering problems. Journal of Algorithms, 53(1):55-84, 2004. Google Scholar
  9. Michel X. Goemans and David P. Williamson. New 3/4-approximation algorithms for the maximum satisfiability problem. SIAM J. Discret. Math., 7(4):656-666, 1994. URL: https://doi.org/10.1137/S0895480192243516.
  10. Michel X. Goemans and David P. Williamson. Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming. J. ACM, 42(6):1115-1145, 1995. URL: https://doi.org/10.1145/227683.227684.
  11. Thekla Hamm, Nina Klobas, George B. Mertzios, and Paul G. Spirakis. The complexity of temporal vertex cover in small-degree graphs. In Thirty-Sixth AAAI Conference on Artificial Intelligence, AAAI 2022, Thirty-Fourth Conference on Innovative Applications of Artificial Intelligence, IAAI 2022, The Twelveth Symposium on Educational Advances in Artificial Intelligence, EAAI 2022 Virtual Event, February 22 - March 1, 2022, pages 10193-10201. AAAI Press, 2022. Google Scholar
  12. Petter Holme. Modern temporal network theory: a colloquium. The European Physical Journal B, 88(9):234, 2015. Google Scholar
  13. Petter Holme and Jari Saramäki. A map of approaches to temporal networks. In Temporal Network Theory, pages 1-24. Springer, 2019. Google Scholar
  14. David Kempe, Jon M. Kleinberg, and Amit Kumar. Connectivity and inference problems for temporal networks. J. Comput. Syst. Sci., 64(4):820-842, 2002. URL: https://doi.org/10.1006/jcss.2002.1829.
  15. Stefan Kratsch and Magnus Wahlström. Representative sets and irrelevant vertices: New tools for kernelization. J. ACM, 67(3):16:1-16:50, 2020. URL: https://doi.org/10.1145/3390887.
  16. Julián Mestre. A primal-dual approximation algorithm for partial vertex cover: Making educated guesses. Algorithmica, 55:227-239, 2009. Google Scholar
  17. 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.
  18. Igor Razgon and Barry O'Sullivan. Almost 2-sat is fixed-parameter tractable. J. Comput. Syst. Sci., 75(8):435-450, 2009. URL: https://doi.org/10.1016/J.JCSS.2009.04.002.
  19. Polina Rozenshtein, Nikolaj Tatti, and Aristides Gionis. The network-untangling problem: from interactions to activity timelines. Data Min. Knowl. Discov., 35(1):213-247, 2021. URL: https://doi.org/10.1007/s10618-020-00717-5.
  20. David P. Williamson and David B. Shmoys. The Design of Approximation Algorithms. Cambridge University Press, 2011. URL: http://www.cambridge.org/de/knowledge/isbn/item5759340/?site_locale=de_DE.
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