The Directed Disjoint Paths Problem with Congestion
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 ≥ 1 and~k ≥ 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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