The Directed Disjoint Paths Problem with Congestion

TL;DR

This paper proves that the directed disjoint paths problem with congestion is NP-complete for many cases, but identifies a specific case (congestion 2 and 3 pairs) that can be solved efficiently.

Contribution

It extends NP-completeness results to broader cases and resolves the complexity of the specific case with congestion 2 and 3 pairs.

Findings

NP-completeness for any constant congestion c ≥ 1 and k ≥ 3c-1 pairs

Refutes the conjecture that congestion 2 cases are polynomial

Shows polynomial-time solvability for congestion 2 and 3 pairs

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.