Skip to main content
SpringerLink
Log in

Timeline Cover in Temporal Graphs: Exact and Approximation Algorithms

  • Conference paper
  • First Online:
Combinatorial Algorithms (IWOCA 2023)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 13889))

Included in the following conference series:

Abstract

In this paper we study a variant of vertex cover on temporal graphs that has been recently introduced for timeline activities summarization in social networks. The problem has been proved to be NP-hard, even in restricted cases. In this paper, we present algorithmic contributions for the problem. First, we present an approximation algorithm of factor \(O(T \log {n})\), on a temporal graph of T timestamps and n vertices. Then, we consider the restriction where at most one temporal edge is defined in each timestamp. For this restriction, which has been recently shown to be NP-hard, we present a \(4(T-1)\) approximation algorithm and a parameterized algorithm when the parameter is the cost (called span) of the solution.

This work was supported by a grant of the Ministry of Research, Innovation and Digitization, CNCS-UEFISCDI, project PN-III-P1-1.1-TE-2021-0253, within PNCDI III.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 64.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 84.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    We recall that the Odd Cycle Transversal problem, given a graph, asks for the removal of the minimum number of edges such that the resulting graph is bipartite.

  2. 2.

    We recall that the Almost 2-SAT, given a formula consisting of clauses on two literals, asks for the removal of the minimum number of clauses so that the resulting formula is satisfiable.

References

  1. Agarwal, A., Charikar, M., Makarychev, K., Makarychev, Y.: O(sqrt(log n)) approximation algorithms for min UnCut, min 2CNF deletion, and directed cut problems. In: Gabow, H.N., Fagin, R. (eds.) Proceedings of the 37th Annual ACM Symposium on Theory of Computing, Baltimore, MD, USA, 22–24 May 2005, pp. 573–581. ACM (2005)

    Google Scholar 

  2. Akrida, E.C., Mertzios, G.B., Spirakis, P.G., Raptopoulos, C.L.: The temporal explorer who returns to the base. J. Comput. Syst. Sci. 120, 179–193 (2021)

    Article  MathSciNet  Google Scholar 

  3. Akrida, E.C., Mertzios, G.B., Spirakis, P.G., Zamaraev, V.: Temporal vertex cover with a sliding time window. J. Comput. Syst. Sci. 107, 108–123 (2020)

    Article  MathSciNet  Google Scholar 

  4. Bafna, V., Berman, P., Fujito, T.: A 2-approximation algorithm for the undirected feedback vertex set problem. SIAM J. Discret. Math. 12(3), 289–297 (1999)

    Article  MathSciNet  Google Scholar 

  5. Bansal, N., Khot, S.: Optimal long code test with one free bit. In: 50th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2009, 25–27 October 2009, Atlanta, Georgia, USA, pp. 453–462. IEEE Computer Society (2009)

    Google Scholar 

  6. Becker, A., Geiger, D.: Optimization of pearl’s method of conditioning and greedy-like approximation algorithms for the vertex feedback set problem. Artif. Intell. 83(1), 167–188 (1996)

    Article  MathSciNet  Google Scholar 

  7. Bumpus, B.M., Meeks, K.: Edge exploration of temporal graphs. In: Flocchini, P., Moura, L. (eds.) IWOCA 2021. LNCS, vol. 12757, pp. 107–121. Springer, Cham (2021). https://doi.org/10.1007/978-3-030-79987-8_8

    Chapter  Google Scholar 

  8. Dondi, R.: Insights into the complexity of disentangling temporal graphs. In: Lago, U.D., Gorla, D. (eds.) Proceedings of the 23th Italian Conference on Theoretical Computer Science, ICTCS 2022. CEUR-WS.org (2022)

    Google Scholar 

  9. Dondi, R., Hosseinzadeh, M.M.: Dense sub-networks discovery in temporal networks. SN Comput. Sci. 2(3), 158 (2021)

    Article  Google Scholar 

  10. Erlebach, T., Hoffmann, M., Kammer, F.: On temporal graph exploration. J. Comput. Syst. Sci. 119, 1–18 (2021)

    Article  MathSciNet  Google Scholar 

  11. Froese, V., Kunz, P., Zschoche, P.: Disentangling the computational complexity of network untangling. In: Raedt, L.D. (ed.) Proceedings of the Thirty-First International Joint Conference on Artificial Intelligence, IJCAI 2022, Vienna, Austria, 23–29 July 2022, pp. 2037–2043. ijcai.org (2022)

    Google Scholar 

  12. Hamm, T., Klobas, N., Mertzios, G.B., Spirakis, P.G.: The complexity of temporal vertex cover in small-degree graphs. CoRR, abs/2204.04832 (2022)

    Google Scholar 

  13. Hochbaum, D.S.: Approximation algorithms for the set covering and vertex cover problems. SIAM J. Comput. 11(3), 555–556 (1982)

    Article  MathSciNet  Google Scholar 

  14. Holme, P.: Modern temporal network theory: a colloquium. Eur. Phys. J. B 88(9), 1–30 (2015). https://doi.org/10.1140/epjb/e2015-60657-4

    Article  Google Scholar 

  15. Holme, P., Saramäki, J.: A map of approaches to temporal networks. In: Holme, P., Saramäki, J. (eds.) Temporal Network Theory. CSS, pp. 1–24. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-23495-9_1

    Chapter  Google Scholar 

  16. Karakostas, G.: A better approximation ratio for the vertex cover problem. ACM Trans. Algorithms 5(4), 41:1–41:8 (2009)

    Google Scholar 

  17. Kempe, D., Kleinberg, J.M., Kumar, A.: Connectivity and inference problems for temporal networks. J. Comput. Syst. Sci. 64(4), 820–842 (2002)

    Article  MathSciNet  Google Scholar 

  18. Marino, A., Silva, A.: Königsberg sightseeing: Eulerian walks in temporal graphs. In: Flocchini, P., Moura, L. (eds.) IWOCA 2021. LNCS, vol. 12757, pp. 485–500. Springer, Cham (2021). https://doi.org/10.1007/978-3-030-79987-8_34

    Chapter  Google Scholar 

  19. Michail, O.: An introduction to temporal graphs: an algorithmic perspective. Internet Math. 12(4), 239–280 (2016)

    Article  MathSciNet  Google Scholar 

  20. Rozenshtein, P., Bonchi, F., Gionis, A., Sozio, M., Tatti, N.: Finding events in temporal networks: segmentation meets densest subgraph discovery. Knowl. Inf. Syst. 62(4), 1611–1639 (2020)

    Article  Google Scholar 

  21. Rozenshtein, P., Tatti, N., Gionis, A.: The network-untangling problem: from interactions to activity timelines. Data Min. Knowl. Discov. 35(1), 213–247 (2021)

    Article  MathSciNet  Google Scholar 

  22. Wu, H., Cheng, J., Huang, S., Ke, Y., Lu, Y., Xu, Y.: Path problems in temporal graphs. Proc. VLDB Endow. 7(9), 721–732 (2014)

    Article  Google Scholar 

  23. Wu, H., Cheng, J., Ke, Y., Huang, S., Huang, Y., Wu, H.: Efficient algorithms for temporal path computation. IEEE Trans. Knowl. Data Eng. 28(11), 2927–2942 (2016)

    Article  Google Scholar 

  24. Zschoche, P., Fluschnik, T., Molter, H., Niedermeier, R.: The complexity of finding small separators in temporal graphs. J. Comput. Syst. Sci. 107, 72–92 (2020)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgement

This work was supported by a grant of the Ministry of Research, Innovation and Digitization, CNCS - UEFISCDI, project number PN-III-P1-1.1-TE-2021-0253, within PNCDI III.

Author information

Authors and Affiliations

  1. Università degli Studi di Bergamo, Bergamo, Italy

    Riccardo Dondi

  2. Department of Computer Science, University of Bucharest, Bucharest, Romania

    Alexandru Popa

Authors
  1. Riccardo Dondi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Alexandru Popa
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Riccardo Dondi .

Editor information

Editors and Affiliations

  1. National Cheng Kung University, Tainan, Taiwan

    Sun-Yuan Hsieh

  2. National Taipei University of Business, Taoyuan, Taiwan

    Ling-Ju Hung

  3. National Taitung University, Taitung, Taiwan

    Chia-Wei Lee

Rights and permissions

Reprints and permissions

Copyright information

© 2023 The Author(s), under exclusive license to Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Dondi, R., Popa, A. (2023). Timeline Cover in Temporal Graphs: Exact and Approximation Algorithms. In: Hsieh, SY., Hung, LJ., Lee, CW. (eds) Combinatorial Algorithms. IWOCA 2023. Lecture Notes in Computer Science, vol 13889. Springer, Cham. https://doi.org/10.1007/978-3-031-34347-6_15

Download citation

  • DOI: https://doi.org/10.1007/978-3-031-34347-6_15

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-031-34346-9

  • Online ISBN: 978-3-031-34347-6

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation