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Timeline Problems in Temporal Graphs: Vertex Cover vs. Dominating Set

Authors Anton Herrmann , Christian Komusiewicz , Nils Morawietz , Frank Sommer

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Subject Classification

ACM Subject Classification
  • Theory of computation → Parameterized complexity and exact algorithms
  • Theory of computation → Graph algorithms analysis
Keywords
  • NP-hard problem
  • FPT-algorithm
  • interval-membership-width
  • Color coding

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Abstract

A temporal graph is a finite sequence of graphs, called snapshots, over the same vertex set. Many temporal graph problems turn out to be much more difficult than their static counterparts. One such problem is Timeline Vertex Cover (also known as MinTimeline_∞), a temporal analogue to the classical Vertex Cover problem. In this problem, one is given a temporal graph 𝒢 and two integers k and 𝓁, and the goal is to cover each edge of each snapshot by selecting for each vertex at most k activity intervals of length at most 𝓁 each. Here, an edge uv in the ith snapshot is covered, if an activity interval of u or v is active at time i. In this work, we continue the algorithmic study of Timeline Vertex Cover and introduce the Timeline Dominating Set problem where we want to dominate all vertices in each snapshot by the selected activity intervals.
We analyze both problems from a classical and parameterized point of view and also consider partial problem versions, where the goal is to cover (dominate) at least t edges (vertices) of the snapshots. With respect to the parameterized complexity, we consider the temporal graph parameters vertex-interval-membership-width (vimw) and interval-membership-width (imw). We show that all considered problems admit FPT-algorithms when parameterized by vimw+k+𝓁. This provides a smaller parameter combination than the ones used for previously known FPT-algorithms for Timeline Vertex Cover. Surprisingly, for imw+k+𝓁, Timeline Dominating Set turns out to be easier than Timeline Vertex Cover, by also admitting an FPT-algorithm, whereas the vertex cover version is NP-hard even if imw+k+𝓁 is constant. We also consider parameterization by combinations of n, the vertex set size, with k or 𝓁 and parameterization by t. Here, we show for example that both partial problems are fixed-parameter tractable for t which significantly improves and generalizes a previous result for a special case of Partial Timeline Vertex Cover with k = 1.

Cite As Get BibTex

Anton Herrmann, Christian Komusiewicz, Nils Morawietz, and Frank Sommer. Timeline Problems in Temporal Graphs: Vertex Cover vs. Dominating Set. In 20th International Symposium on Parameterized and Exact Computation (IPEC 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 358, pp. 12:1-12:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.IPEC.2025.12

Author Details

Anton Herrmann
  • Algorithmics and Computational Complexity, Technische Universität Berlin, Germany
Christian Komusiewicz
  • Institute of Computer Science, Friedrich Schiller University Jena, Germany
Nils Morawietz
  • Institute of Computer Science, Friedrich Schiller University Jena, Germany
  • LaBRI, Université de Bordeaux, Talence, France
Frank Sommer
  • Institute of Logic and Computation, TU Wien, Austria
  • Institute of Computer Science, Friedrich Schiller University Jena, Germany

Funding

  • Morawietz, Nils: Supported by the French ANR, project ANR-22-CE48-0001 (TEMPOGRAL).
  • Sommer, Frank: Supported by the Alexander von Humboldt Foundation.

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