The Complexity of Computing Optimum Labelings for Temporal Connectivity

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

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

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).

Original languageEnglish
Title of host publication47th International Symposium on Mathematical Foundations of Computer Science, MFCS 2022
EditorsStefan Szeider, Robert Ganian, Alexandra Silva
PublisherSchloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
ISBN (Electronic)9783959772563
DOIs
StatePublished - 1 Aug 2022
Event47th International Symposium on Mathematical Foundations of Computer Science, MFCS 2022 - Vienna, Austria
Duration: 22 Aug 202226 Aug 2022

Publication series

NameLeibniz International Proceedings in Informatics, LIPIcs
Volume241
ISSN (Print)1868-8969

Conference

Conference47th International Symposium on Mathematical Foundations of Computer Science, MFCS 2022
Country/TerritoryAustria
CityVienna
Period22/08/2226/08/22

Keywords

  • Steiner Tree
  • Temporal graph
  • foremost temporal path
  • graph labeling
  • temporal connectivity

ASJC Scopus subject areas

  • Software

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