### Abstract

Given a set of directed paths (called lines) L, a public transportation network is a directed graph G (L) = (V (L) , A (L) ) which contains exactly the vertices and arcs of every line l a L. An st-route is a pair (pi, gamma) where gamma = aOE (c) l (1),aEuro broken vertical bar, l (h) > is a line sequence and pi is an st-path in G (L) which is the concatenation of subpaths of the lines l (1),aEuro broken vertical bar, l (h) , in this order. Given a threshold beta, we present an algorithm for listing all st-paths pi for which a route (pi, gamma) with |gamma| ae beta exists, and we show that the running time of this algorithm is polynomial with respect to the input and the output size. We also present an algorithm for listing all line sequences gamma with |gamma| ae beta for which a route (pi, gamma) exists, and show how to speed it up using preprocessing. Moreover, we show that for the problem of finding an st-route (pi, gamma) that minimizes the number of different lines in gamma, even computing an -approximation is NP-hard.

Original language | English |
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Pages (from-to) | 600-621 |

Number of pages | 22 |

Journal | Theory of Computing Systems |

Volume | 62 |

Issue number | 3 |

DOIs | |

Publication status | Published - 1 Apr 2018 |

### Keywords

- K SHORTEST PATHS
- Listing algorithm
- NP-hardness
- Public transportation

## Cite this

*Theory of Computing Systems*,

*62*(3), 600-621. https://doi.org/10.1007/s00224-016-9747-4