The Directed Disjoint Paths Problem with Congestion
Discrete Mathematics
2025-07-17 v1 Combinatorics
Abstract
The classic result by Fortune, Hopcroft, and Wyllie [TCS~'80] states that the directed disjoint paths problem is NP-complete even for two pairs of terminals. Extending this well-known result, we show that the directed disjoint paths problem is NP-complete for any constant congestion and~ pairs of terminals. This refutes a conjecture by Giannopoulou et al. [SODA~'22], which says that the directed disjoint paths problem with congestion two is polynomial-time solvable for any constant number of terminal pairs. We then consider the cases that are not covered by this hardness result. The first nontrivial case is and . Our second main result is to show that this case is polynomial-time solvable.
Cite
@article{arxiv.2507.12096,
title = {The Directed Disjoint Paths Problem with Congestion},
author = {Matthias Bentert and Dario Cavallaro and Amelie Heindl and Ken-ichi Kawarabayashi and Stephan Kreutzer and Johannes Schröder},
journal= {arXiv preprint arXiv:2507.12096},
year = {2025}
}