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arXiv · 2507.12096

The Directed Disjoint Paths Problem with Congestion

Matthias Bentert·Dario Cavallaro·Amelie Heindl·Ken-ichi Kawarabayashi·Stephan Kreutzer·Johannes Schröder
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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 $c \geq 1$ and~$k \geq 3c-1$ 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 $k$ of terminal pairs. We then consider the cases that are not covered by this hardness result. The first nontrivial case is $c=2$ and $k = 3$. Our second main result is to show that this case is polynomial-time solvable.

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Matthias Bentert, Dario Cavallaro, Amelie Heindl, Ken-ichi Kawarabayashi, Stephan Kreutzer, Johannes Schröder. 2025-07-16. The Directed Disjoint Paths Problem with Congestion. https://arxiv.org/abs/2507.12096

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