## Abstract

Computing a (short) path between two vertices is one of the most fundamental primitives in graph algorithmics. In recent years, the study of paths in temporal graphs, that is, graphs where the vertex set is fixed but the edge set changes over time, gained more and more attention. A path is time-respecting, or temporal, if it uses edges with non-decreasing time stamps. We investigate a basic constraint for temporal paths, where the time spent at each vertex must not exceed a given duration ∆, referred to as ∆-restless temporal paths. This constraint arises naturally in the modeling of real-world processes like packet routing in communication networks and infection transmission routes of diseases where recovery confers lasting resistance. While finding temporal paths without waiting time restrictions is known to be doable in polynomial time, we show that the “restless variant” of this problem becomes computationally hard even in very restrictive settings. For example, it is W[1]-hard when parameterized by the feedback vertex number or the pathwidth of the underlying graph. The main question thus is whether the problem becomes tractable in some natural settings. We explore several natural parameterizations, presenting FPT algorithms for three kinds of parameters: (1) output-related parameters (here, the maximum length of the path), (2) classical parameters applied to the underlying graph (e.g., feedback edge number), and (3) a new parameter called timed feedback vertex number, which captures finer-grained temporal features of the input temporal graph, and which may be of interest beyond this work.

Original language | English |
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Title of host publication | 31st International Symposium on Algorithms and Computation, ISAAC 2020 |

Editors | Yixin Cao, Siu-Wing Cheng, Minming Li |

Publisher | Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing |

Pages | 301-3018 |

Number of pages | 2718 |

ISBN (Electronic) | 9783959771733 |

DOIs | |

State | Published - 1 Dec 2020 |

Externally published | Yes |

Event | 31st International Symposium on Algorithms and Computation, ISAAC 2020 - Virtual, Hong Kong, China Duration: 14 Dec 2020 → 18 Dec 2020 |

### Publication series

Name | Leibniz International Proceedings in Informatics, LIPIcs |
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Volume | 181 |

ISSN (Print) | 1868-8969 |

### Conference

Conference | 31st International Symposium on Algorithms and Computation, ISAAC 2020 |
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Country/Territory | China |

City | Virtual, Hong Kong |

Period | 14/12/20 → 18/12/20 |

## Keywords

- Disease spreading
- NP-hard problems
- Parameterized algorithms
- Restless temporal paths
- Temporal graphs
- Timed feedback vertex set
- Waiting-time policies

## ASJC Scopus subject areas

- Software