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Temporal Graph Realization with Bounded Stretch

Authors George B. Mertzios , Hendrik Molter , Nils Morawietz , Paul G. Spirakis

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ACM Subject Classification
  • Theory of computation → Graph algorithms analysis
  • Mathematics of computing → Discrete mathematics
Keywords
  • Temporal graph
  • periodic temporal labeling
  • fastest temporal path
  • graph realization
  • temporal connectivity
  • stretch

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Abstract

A periodic temporal graph, in its simplest form, is a graph in which every edge appears exactly once in the first Δ time steps, and then it reappears recurrently every Δ time steps, where Δ is a given period length. This model offers a natural abstraction of transportation networks where each transportation link connects two destinations periodically. From a network design perspective, a crucial task is to assign the time-labels on the edges in a way that optimizes some criterion. In this paper we introduce a very natural optimality criterion that captures how the temporal distances of all vertex pairs are "stretched", compared to their physical distances, i.e. their distances in the underlying static (non-temporal) graph. Given a static graph G, the task is to assign to each edge one time-label between 1 and Δ such that, in the resulting periodic temporal graph with period Δ, the duration of the fastest temporal path from any vertex u to any other vertex v is at most α times the distance between u and v in G. Here, the value of α measures how much the shortest paths are allowed to be stretched once we assign the periodic time-labels. 
Our results span three different directions: First, we provide a series of approximation and NP-hardness results. Second, we provide approximation and fixed-parameter algorithms. Among them, we provide a simple polynomial-time algorithm (the radius-algorithm) which always guarantees an approximation strictly smaller than Δ, and which also computes the optimum stretch in some cases. Third, we consider a parameterized local search extension of the problem where we are given the temporal labeling of the graph, but we are allowed to change the time-labels of at most k edges; for this problem we prove that it is W[2]-hard but admits an XP algorithm with respect to k.

Cite As Get BibTex

George B. Mertzios, Hendrik Molter, Nils Morawietz, and Paul G. Spirakis. Temporal Graph Realization with Bounded Stretch. In 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 345, pp. 75:1-75:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.MFCS.2025.75

Author Details

George B. Mertzios
  • Department of Computer Science, Durham University, UK
Hendrik Molter
  • Department of Computer Science, Ben-Gurion University of the Negev, Beer-Sheva, Israel
Nils Morawietz
  • LaBRI, Université de Bordeaux, France, Institute of Computer Science, Friedrich Schiller University Jena, Germany
Paul G. Spirakis
  • Department of Computer Science, University of Liverpool, UK

Funding

  • Mertzios, George B.: Supported by the EPSRC grant EP/P020372/1.
  • Molter, Hendrik: Supported by the Israel Science Foundation, grant nr. 1470/24, by the European Union’s Horizon Europe research and innovation programme under grant agreement 949707, and by the European Research Council, grant nr. 101039913 (PARAPATH).
  • Morawietz, Nils: Supported by the French ANR, project ANR-22-CE48-0001 (TEMPOGRAL)
  • Spirakis, Paul G.: Supported by the EPSRC grant EP/P02002X/1.

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