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Minimize the Sum of Waiting Times in Periodic Temporal Trees

Authors Julia Meusel , Nils Morawietz , Matthias Müller-Hannemann , Klaus Reinhardt

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Subject Classification

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

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Abstract

We introduce and analyze the problem of finding a Δ-labeling λ for an undirected tree G = (V,E), such that the sum of overall waiting times of fastest paths between all vertex pairs is minimized in the Δ-periodic temporal graph (G,λ). That is, we aim to minimize ∑_{(u,v) ∈ V×V} (dur(u,v)-dist(u,v)), where dur(u,v) is the duration of a fastest temporal path from u to v and dist(u,v) is the length of the shortest path between u and v in G. We show that this objective function essentially boils down to a known problem about partitioning a set of natural numbers that has applications in scheduling. From that problem we lift and adapt several upper and lower bounds for our problem. For example, we show that the problem admits an EPTAS, that is, an algorithm that can compute a (1+ε)-approximation to our problem in time f(1/ε) ⋅ n^𝒪(1) for each ε > 0. To the best of our knowledge, this is the first example of an efficient approximation algorithm for a temporal graph realization problem.

Cite As Get BibTex

Julia Meusel, Nils Morawietz, Matthias Müller-Hannemann, and Klaus Reinhardt. Minimize the Sum of Waiting Times in Periodic Temporal Trees. In 5th Symposium on Algorithmic Foundations of Dynamic Networks (SAND 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 373, pp. 19:1-19:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.SAND.2026.19

Author Details

Julia Meusel
  • Martin Luther University Halle-Wittenberg, Germany
Nils Morawietz
  • LaBRI, Université de Bordeaux, Talence, France
  • Institute of Computer Science, Friedrich Schiller University Jena, Germany
Matthias Müller-Hannemann
  • Martin Luther University Halle-Wittenberg, Germany
Klaus Reinhardt
  • Martin Luther University Halle-Wittenberg, Germany

Funding

  • Morawietz, Nils: Supported by the French ANR, project ANR-22-CE48-0001 (TEMPOGRAL).

References

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