The Directed Disjoint Paths Problem with Congestion

2025-07-16

AI Summary

This paper investigates the Directed Congestion-bounded Disjoint Paths problem (Dir-CDP): given a directed graph, $k$ terminal pairs, and a congestion bound $c$, decide whether there exist $k$ vertex-disjoint paths such that each edge is used by at most $c$ paths. Employing constructive reductions combined with dynamic programming, we establish that Dir-CDP is NP-complete for all $c geq 1$ and $k geq 3c - 1$, thereby refuting the conjecture that the problem is polynomial-time solvable when $c = 2$. Concurrently, we devise a polynomial-time algorithm for the case $c = 2$, $k = 3$, precisely delineating the complexity threshold within this parameter regime. Our approach uniformly handles arbitrary constant congestion bounds and uncovers the critical interplay between $k$ and $c$, resolving a fundamental open question and filling a key gap in the complexity landscape of the directed variant.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Combinatorial Optimization
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Security and Privacy: Large-scale security measurements

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.

Problem

• NP-completeness of directed disjoint paths with congestion
• Refuting conjecture on polynomial-time solvability
• Polynomial-time solution for specific congestion case

Innovation

• Proving NP-completeness for constant congestion
• Refuting conjecture on polynomial-time solvability
• Solving polynomial-time for c=2 and k=3