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Temporal Separators with Deadlines

Authors Hovhannes A. Harutyunyan, Kamran Koupayi, Denis Pankratov

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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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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.

Cite As Get BibTex

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

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

Funding

The research presented in this work is supported by NSERC.

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