The Complexity of Transitively Orienting Temporal Graphs

George B. Mertzios, Hendrik Molter, Malte Renken, Paul G. Spirakis, Philipp Zschoche

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

Abstract

In a temporal network with discrete time-labels on its edges, entities and information can only "flow"along sequences of edges whose time-labels are non-decreasing (resp. increasing), i.e. along temporal (resp. strict temporal) paths. Nevertheless, in the model for temporal networks of [Kempe, Kleinberg, Kumar, JCSS, 2002], the individual time-labeled edges remain undirected: an edge e = {u, v} with time-label t specifies that "u communicates with v at time t". This is a symmetric relation between u and v, and it can be interpreted that the information can flow in either direction. In this paper we make a first attempt to understand how the direction of information flow on one edge can impact the direction of information flow on other edges. More specifically, naturally extending the classical notion of a transitive orientation in static graphs, we introduce the fundamental notion of a temporal transitive orientation and we systematically investigate its algorithmic behavior in various situations. An orientation of a temporal graph is called temporally transitive if, whenever u has a directed edge towards v with time-label t1 and v has a directed edge towards w with time-label t2 ≥ t1, then u also has a directed edge towards w with some time-label t3 ≥ t2. If we just demand that this implication holds whenever t2 > t1, the orientation is called strictly temporally transitive, as it is based on the fact that there is a strict directed temporal path from u to w. Our main result is a conceptually simple, yet technically quite involved, polynomial-time algorithm for recognizing whether a given temporal graph G is transitively orientable. In wide contrast we prove that, surprisingly, it is NP-hard to recognize whether G is strictly transitively orientable. Additionally we introduce and investigate further related problems to temporal transitivity, notably among them the temporal transitive completion problem, for which we prove both algorithmic and hardness results.

Original languageEnglish
Title of host publication46th International Symposium on Mathematical Foundations of Computer Science, MFCS 2021
EditorsFilippo Bonchi, Simon J. Puglisi
PublisherSchloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
ISBN (Electronic)9783959772013
DOIs
StatePublished - 1 Aug 2021
Event46th International Symposium on Mathematical Foundations of Computer Science, MFCS 2021 - Tallinn, Estonia
Duration: 23 Aug 202127 Aug 2021

Publication series

NameLeibniz International Proceedings in Informatics, LIPIcs
Volume202
ISSN (Print)1868-8969

Conference

Conference46th International Symposium on Mathematical Foundations of Computer Science, MFCS 2021
Country/TerritoryEstonia
CityTallinn
Period23/08/2127/08/21

Keywords

  • Np-hardness
  • Polynomial-time algorithm
  • Satisfiability
  • Temporal graph
  • Transitive closure
  • Transitive orientation

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