Klobas, N. and Mertzios, G.B. and Molter, H. and Spirakis, P.G. (2022) 'The complexity of computing optimum labelings for temporal connectivity.', *47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022)* Vienna, Austria, 22-26 Aug 2022.

## 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).

Item Type: | Conference item (Paper) |
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Full text: | (VoR) Version of Record Available under License - Creative Commons Attribution 4.0. Download PDF (1174Kb) |

Status: | Peer-reviewed |

Publisher Web site: | https://doi.org/10.4230/LIPIcs.MFCS.2022.14 |

Publisher statement: | © Nina Klobas, George B. Mertzios, Hendrik Molter, and Paul G. Spirakis; licensed under Creative Commons License CC-BY 4.0 |

Date accepted: | 21 June 2022 |

Date deposited: | 26 July 2022 |

Date of first online publication: | 2022 |

Date first made open access: | 26 July 2022 |

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