The Directed Disjoint Paths Problem with Congestion

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 $c \ge 1$ and $k \ge 3c - 1$ 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.