On the Parameterized Tractability of Packing Vertex-Disjoint A-Paths with Length Constraints

2026-02-16

AI Summary

This study investigates the parameterized complexity of the (A,ℓ)-path packing problem in undirected graphs, which asks whether there exist k vertex-disjoint paths each of length exactly ℓ and with both endpoints in a specified vertex set A. By leveraging parameterized complexity theory, kernelization techniques, and graph decomposition methods, the work provides the first comprehensive complexity characterization under multiple structural graph parameters. Specifically, it establishes W[1]-hardness when parameterized by the distance to pathwidth plus |A| (dtp + |A|), designs fixed-parameter tractable (FPT) algorithms for the combined parameters cvd + |A| and cvd + ℓ (where cvd denotes the distance to cluster), and constructs a polynomial kernel of size O(vc²) parameterized by the vertex cover number vc.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Distributed CSP/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
• 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

Given an undirected graph G and a set A \subseteq V(G), an A-path is a path in G that starts and ends at two distinct vertices of A with intermediate vertices in V(G) \setminus A. An A-path is called an (A,\ell)-path if the length of the path is exactly \ell. In the {\sc (A, \ell)-Path Packing} problem (ALPP), we seek to determine whether there exist k vertex-disjoint (A, \ell)-paths in G or not. We pursue this problem with respect to structural parameters. We prove that ALPP is W[1]-hard when it is parameterized by the combined parameter distance to path (dtp) and |A|. In addition, we consider the combined parameters distance to cluster (cvd) + |A| and distance to cluster (cvd) + \ell. For both these combined parameters, we provide FPT algorithms. Finally, we consider the vertex cover number (vc) as the parameter and provide a kernel with O(vc^2) vertices.

Problem

• A-path
• length constraints
• vertex-disjoint paths
• parameterized complexity
• path packing

Innovation

• parameterized complexity
• A-path packing
• FPT algorithm
• kernelization
• structural parameters