Temporal Separators with Deadlines

Authors Hovhannes A. Harutyunyan, Kamran Koupayi, Denis Pankratov



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Author Details

Hovhannes A. Harutyunyan
  • Department of Computer Science and Software Engineering, Concordia University, Montreal, Canada
Kamran Koupayi
  • Department of Computer Science and Software Engineering, Concordia University, Montreal, Canada
Denis Pankratov
  • Department of Computer Science and Software Engineering, Concordia University, Montreal, Canada

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Hovhannes A. Harutyunyan, Kamran Koupayi, and Denis Pankratov. Temporal Separators with Deadlines. In 34th International Symposium on Algorithms and Computation (ISAAC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 283, pp. 38:1-38:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)
https://doi.org/10.4230/LIPIcs.ISAAC.2023.38

Abstract

We study temporal analogues of the Unrestricted Vertex Separator problem from the static world. An (s,z)-temporal separator is a set of vertices whose removal disconnects vertex s from vertex z for every time step in a temporal graph. The (s,z)-Temporal Separator problem asks to find the minimum size of an (s,z)-temporal separator for the given temporal graph. The (s,z)-Temporal Separator problem is known to be NP-hard in general, although some special cases (such as bounded treewidth) admit efficient algorithms [Fluschnik et al., 2020]. We introduce a generalization of this problem called the (s,z,t)-Temporal Separator problem, where the goal is to find a smallest subset of vertices whose removal eliminates all temporal paths from s to z which take less than t time steps. Let τ denote the number of time steps over which the temporal graph is defined (we consider discrete time steps). We characterize the set of parameters τ and t when the problem is NP-hard and when it is polynomial time solvable. Then we present a τ-approximation algorithm for the (s,z)-Temporal Separator problem and convert it to a τ²-approximation algorithm for the (s,z,t)-Temporal Separator problem. We also present an inapproximability lower bound of Ω(ln(n) + ln(τ)) for the (s,z,t)-Temporal Separator problem assuming that NP ⊄ DTIME(n^{log log n}). Then we consider three special families of graphs: (1) graphs of branchwidth at most 2, (2) graphs G such that the removal of s and z leaves a tree, and (3) graphs of bounded pathwidth. We present polynomial-time algorithms to find a minimum (s,z,t)-temporal separator for (1) and (2). As for (3), we show a polynomial-time reduction from the Discrete Segment Covering problem with bounded-length segments to the (s,z,t)-Temporal Separator problem where the temporal graph has bounded pathwidth.

Subject Classification

ACM Subject Classification
  • Theory of computation → Dynamic graph algorithms
Keywords
  • Temporal graphs
  • dynamic graphs
  • vertex separator
  • vertex cut
  • separating set
  • deadlines
  • inapproximability
  • approximation algorithms

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References

  1. Susanne Albers. Online algorithms: a survey. Mathematical Programming, 97(1-2):3-26, 2003. Google Scholar
  2. Haeder Y. Althoby, Mohamed Didi Biha, and André Sesboüé. Exact and heuristic methods for the vertex separator problem. Computers and Industrial Engineering, 139:106135, 2020. URL: https://doi.org/10.1016/j.cie.2019.106135.
  3. Aris Anagnostopoulos, Ravi Kumar, Mohammad Mahdian, Eli Upfal, and Fabio Vandin. Algorithms on evolving graphs. In Proceedings of the 3rd Innovations in Theoretical Computer Science Conference, pages 149-160, 2012. Google Scholar
  4. Dan Bergren, Eduard Eiben, Robert Ganian, and Iyad Kanj. On covering segments with unit intervals. In 37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020). Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2020. Google Scholar
  5. Hans L. Bodlaender and Dimitrios M. Thilikos. Constructive linear time algorithms for branchwidth. In Pierpaolo Degano, Roberto Gorrieri, and Alberto Marchetti-Spaccamela, editors, Automata, Languages and Programming, pages 627-637, Berlin, Heidelberg, 1997. Springer Berlin Heidelberg. Google Scholar
  6. 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
  7. Thomas H Cormen, Charles E Leiserson, Ronald L Rivest, and Clifford Stein. Introduction to algorithms. MIT press, 2009. Google Scholar
  8. Xiaojie Deng, Bingkai Lin, and Chihao Zhang. Multi-multiway cut problem on graphs of bounded branch width. In Frontiers in Algorithmics and Algorithmic Aspects in Information and Management, pages 315-324. Springer, 2013. Google Scholar
  9. Jessica Enright, Kitty Meeks, George B Mertzios, and Viktor Zamaraev. Deleting edges to restrict the size of an epidemic in temporal networks. arXiv preprint, 2018. URL: https://arxiv.org/abs/1805.06836.
  10. Uriel Feige. A threshold of ln n for approximating set cover. Journal of the ACM (JACM), 45(4):634-652, 1998. Google Scholar
  11. Joan Feigenbaum, Sampath Kannan, Andrew McGregor, Siddharth Suri, and Jian Zhang. On graph problems in a semi-streaming model. In International Colloquium on Automata, Languages, and Programming, pages 531-543. Springer, 2004. Google Scholar
  12. Joan Feigenbaum, Sampath Kannan, Andrew McGregor, Siddharth Suri, and Jian Zhang. On graph problems in a semi-streaming model. Departmental Papers (CIS), page 236, 2005. Google Scholar
  13. Afonso Ferreira. Building a reference combinatorial model for manets. IEEE network, 18(5):24-29, 2004. Google Scholar
  14. Paola Flocchini, Bernard Mans, and Nicola Santoro. Exploration of periodically varying graphs. In International Symposium on Algorithms and Computation, pages 534-543. Springer, 2009. Google Scholar
  15. 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
  16. Naveen Garg, Vijay V Vazirani, and Mihalis Yannakakis. Multiway cuts in directed and node weighted graphs. In International Colloquium on Automata, Languages, and Programming, pages 487-498. Springer, 1994. Google Scholar
  17. Hovhannes A. Harutyunyan, Kamran Koupayi, and Denis Pankratov. Temporal separators with deadlines, 2023. URL: https://arxiv.org/abs/2309.14185.
  18. David Kempe, Jon Kleinberg, and Amit Kumar. Connectivity and inference problems for temporal networks. Journal of Computer and System Sciences, 64(4):820-842, 2002. Google Scholar
  19. George B Mertzios, Othon Michail, Ioannis Chatzigiannakis, and Paul G Spirakis. Temporal network optimization subject to connectivity constraints. In International Colloquium on Automata, Languages, and Programming, pages 657-668. Springer, 2013. Google Scholar
  20. Othon Michail. An introduction to temporal graphs: An algorithmic perspective. Internet Mathematics, 12(4):239-280, 2016. Google Scholar
  21. Neil Robertson and Paul D Seymour. Graph minors. x. obstructions to tree-decomposition. Journal of Combinatorial Theory, Series B, 52(2):153-190, 1991. Google Scholar
  22. Neil Robertson and P.D. Seymour. Graph minors. i. excluding a forest. Journal of Combinatorial Theory, Series B, 35(1):39-61, 1983. URL: https://doi.org/10.1016/0095-8956(83)90079-5.
  23. Ryan A Rossi, Brian Gallagher, Jennifer Neville, and Keith Henderson. Modeling dynamic behavior in large evolving graphs. In Proceedings of the sixth ACM international conference on Web search and data mining, pages 667-676, 2013. Google Scholar
  24. Daniel D Sleator and Robert Endre Tarjan. A data structure for dynamic trees. Journal of computer and system sciences, 26(3):362-391, 1983. Google Scholar
  25. Huanhuan Wu, James Cheng, Silu Huang, Yiping Ke, Yi Lu, and Yanyan Xu. Path problems in temporal graphs. Proceedings of the VLDB Endowment, 7(9):721-732, 2014. Google Scholar
  26. 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
  27. 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. Google Scholar
  28. Philipp Zschoche, Till Fluschnik, Hendrik Molter, and Rolf Niedermeier. The complexity of finding small separators in temporal graphs. Journal of Computer and System Sciences, 107:72-92, 2020. Google Scholar