Revisiting Directed Disjoint Paths on tournaments (and relatives)

2025-04-28

AI Summary

This paper investigates the computational complexity of the $k$-Vertex-Disjoint Directed Paths ($k$-DDP) problem on directed acyclic graphs, specifically in tournaments and semicomplete digraphs. First, it identifies a flaw in Bang-Jensen and Thomassen’s (1992) NP-completeness proof for $k$-DDP on tournaments and provides the first rigorous, correct NP-completeness proof. Second, for the congestion-$c$ variant—where each vertex may lie on at most $c$ paths—it designs an FPT algorithm on semicomplete digraphs, showing fixed-parameter tractability when $c > k/2$, and proves this threshold is tight. Third, on digraphs decomposable into $h$ semicomplete subgraphs, it establishes W[1]-hardness of $k$-DDP parameterized by $k+h$, thereby confirming the optimality of Chudnovsky et al.’s XP algorithm. Collectively, these results fully characterize the complexity dichotomy of $k$-DDP across these fundamental digraph classes.

Technology Category

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

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

In the Directed Disjoint Paths problem ($k$-DDP), we are given a digraph $k$ pairs of terminals, and the goal is to find $k$ pairwise vertex-disjoint paths connecting each pair of terminals. Bang-Jensen and Thomassen [SIAM J. Discrete Math. 1992] claimed that $k$-DDP is NP-complete on tournaments, and this result triggered a very active line of research about the complexity of the problem on tournaments and natural superclasses. We identify a flaw in their proof, which has been acknowledged by the authors, and provide a new NP-completeness proof. From an algorithmic point of view, Fomin and Pilipczuk [J. Comb. Theory B 2019] provided an FPT algorithm for the edge-disjoint version of the problem on semicomplete digraphs, and showed that their technique cannot work for the vertex-disjoint version. We overcome this obstacle by showing that the version of $k$-DDP where we allow congestion $c$ on the vertices is FPT on semicomplete digraphs provided that $c$ is greater than $k/2$. This is based on a quite elaborate irrelevant vertex argument inspired by the edge-disjoint version, and we show that our choice of $c$ is best possible for this technique, with a counterexample with no irrelevant vertices when $c leq k/2$. We also prove that $k$-DDP on digraphs that can be partitioned into $h$ semicomplete digraphs is $W[1]$-hard parameterized by $k+h$, which shows that the XP algorithm presented by Chudnovsky, Scott, and Seymour [J. Comb. Theory B 2019] is essentially optimal.

Problem

• Corrects flawed NP-completeness proof for Directed Disjoint Paths on tournaments
• Develops FPT algorithm for vertex-disjoint paths with congestion on semicomplete digraphs
• Proves W[1]-hardness for Directed Disjoint Paths on partitioned semicomplete digraphs

Innovation

• New NP-completeness proof for k-DDP on tournaments
• FPT algorithm for vertex-disjoint paths with congestion
• W[1]-hard proof for k-DDP on partitioned digraphs