### Abstract

We study the design of small cost temporally connected graphs, under various constraints. We mainly consider undirected graphs of n vertices, where each edge has an associated set of discrete availability instances (labels). A journey from vertex u to vertex v is a path from u to v where successive path edges have strictly increasing labels. A graph is temporally connected iff there is a (u, v)-journey for any pair of vertices u, v, u ≠ v. We first give a simple polynomial-time algorithm to check whether a given temporal graph is temporally connected. We then consider the case in which a designer of temporal graphs can freely choose availability instances for all edges and aims for temporal connectivity with very small cost; the cost is the total number of availability instances used. We achieve this via a simple polynomial-time procedure which derives designs of cost linear in n, and at most the optimal cost plus 2. To show this, we prove a lower bound on the cost for any undirected graph. However, there are pragmatic cases where one is not free to design a temporally connected graph anew, but is instead given a temporal graph design with the claim that it is temporally connected, and wishes to make it more cost-efficient by removing labels without destroying temporal connectivity (redundant labels). Our main technical result is that computing the maximum number of redundant labels is APX-hard, i.e., there is no PTAS unless P = NP. On the positive side, we show that in dense graphs with random edge availabilities, all but Θ(n) labels are redundant whp. A temporal design may, however, be minimal, i.e., no redundant labels exist. We show the existence of minimal temporal designs with at least n log n labels.

Original language | English (US) |
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Title of host publication | Approximation and Online Algorithms - 13th International Workshop, WAOA 2015, Revised Selected Papers |

Editors | Martin Skutella, Laura Sanità |

Publisher | Springer Verlag |

Pages | 84-96 |

Number of pages | 13 |

ISBN (Print) | 9783319286839 |

DOIs | |

State | Published - Jan 1 2015 |

Event | 13th International Workshop on Approximation and Online Algorithms, WAOA 2015 - Patras, Greece Duration: Sep 17 2015 → Sep 18 2015 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
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Volume | 9499 |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Conference

Conference | 13th International Workshop on Approximation and Online Algorithms, WAOA 2015 |
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Country | Greece |

City | Patras |

Period | 9/17/15 → 9/18/15 |

### ASJC Scopus subject areas

- Theoretical Computer Science
- Computer Science(all)

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## Cite this

*Approximation and Online Algorithms - 13th International Workshop, WAOA 2015, Revised Selected Papers*(pp. 84-96). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 9499). Springer Verlag. https://doi.org/10.1007/978-3-319-28684-6_8