## Abstract

We study the computational complexity of routing multiple objects through a network while avoiding collision: Given a graph G with two distinct terminals and two positive integers p,k, the question is whether one can connect the terminals by at least p routes (walks, trails, paths; the latter two without repeated edges or vertices, respectively) such that in at most k edges it happens that we traverse them in more than one route at the same time. We prove that for paths and trails the problem is NP-hard on undirected and directed planar graphs even if the maximum vertex degree or k≥0 is constant. For walks we prove polynomial-time solvability on undirected graphs for unbounded k and on directed graphs if k≥0 is constant. We additionally study variants where the maximum length of a route is restricted. For walks this variant becomes NP-hard on undirected graphs.

Original language | English |
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Pages (from-to) | 69-86 |

Number of pages | 18 |

Journal | Journal of Computer and System Sciences |

Volume | 102 |

DOIs | |

State | Published - 1 Jun 2019 |

## Keywords

- Dynamic flows
- Many-one reduction
- NP-hardness
- Shared edges

## ASJC Scopus subject areas

- Theoretical Computer Science
- Computer Networks and Communications
- Computational Theory and Mathematics
- Applied Mathematics