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The Complexity of Computing Optimum Labelings for Temporal Connectivity

Authors Nina Klobas , George B. Mertzios , Hendrik Molter , Paul G. Spirakis

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Author Details

Nina Klobas
  • Department of Computer Science, Durham University, UK
George B. Mertzios
  • Department of Computer Science, Durham University, UK
Hendrik Molter
  • Department of Industrial Engineering and Management, Ben-Gurion University of the Negev, Israel
Paul G. Spirakis
  • Department of Computer Science, University of Liverpool, UK
  • Computer Engineering & Informatics Department, University of Patras, Greece

Funding

  • Mertzios, George B.: Supported by the EPSRC grant EP/P020372/1.
  • Molter, Hendrik: Supported by the ISF, grant No. 1070/20.
  • Spirakis, Paul G.: Supported by the NeST initiative of the School of EEE and CS at the University of Liverpool and by the EPSRC grant EP/P02002X/1.

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Nina Klobas, George B. Mertzios, Hendrik Molter, and Paul G. Spirakis. The Complexity of Computing Optimum Labelings for Temporal Connectivity. In 47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 241, pp. 62:1-62:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022) https://doi.org/10.4230/LIPIcs.MFCS.2022.62

Abstract

A graph is temporally connected if there exists a strict temporal path, i.e., a path whose edges have strictly increasing labels, from every vertex u to every other vertex v. In this paper we study temporal design problems for undirected temporally connected graphs. The basic setting of these optimization problems is as follows: given a connected undirected graph G, what is the smallest number |λ| of time-labels that we need to add to the edges of G such that the resulting temporal graph (G,λ) is temporally connected? As it turns out, this basic problem, called Minimum Labeling (ML), can be optimally solved in polynomial time. However, exploiting the temporal dimension, the problem becomes more interesting and meaningful in its following variations, which we investigate in this paper. First we consider the problem Min. Aged Labeling (MAL) of temporally connecting the graph when we are given an upper-bound on the allowed age (i.e., maximum label) of the obtained temporal graph (G,λ). Second we consider the problem Min. Steiner Labeling (MSL), where the aim is now to have a temporal path between any pair of "important" vertices which lie in a subset R ⊆ V, which we call the terminals. This relaxed problem resembles the problem Steiner Tree in static (i.e., non-temporal) graphs. However, due to the requirement of strictly increasing labels in a temporal path, Steiner Tree is not a special case of MSL. Finally we consider the age-restricted version of MSL, namely Min. Aged Steiner Labeling (MASL). Our main results are threefold: we prove that (i) MAL becomes NP-complete on undirected graphs, while (ii) MASL becomes W[1]-hard with respect to the number |R| of terminals. On the other hand we prove that (iii) although the age-unrestricted problem MSL remains NP-hard, it is in FPT with respect to the number |R| of terminals. That is, adding the age restriction, makes the above problems strictly harder (unless P=NP or W[1]=FPT).

Subject Classification

ACM Subject Classification
  • Theory of computation → Graph algorithms analysis
  • Mathematics of computing → Discrete mathematics
Keywords
  • Temporal graph
  • graph labeling
  • foremost temporal path
  • temporal connectivity
  • Steiner Tree

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