Parameterized Complexity and Approximability of the Maximum Node-Disjoint Paths Problem

Conceitos Básicos

The Maximum Node-Disjoint Paths problem is intractable under various structural parameterizations, but can be efficiently approximated for some of these parameters using FPT approximation schemes.

Resumo

The paper revisits the Maximum Node-Disjoint Paths (MaxNDP) problem, which is the optimization version of the famous Node-Disjoint Paths problem. In this problem, given a graph G, a set of k demand pairs, and an integer ℓ, the goal is to determine the maximum number of vertex-disjoint paths that can be used to route at least ℓ of the demand pairs.
The authors present several results that improve and clarify the state of the art regarding the parameterized complexity and approximability of MaxNDP:

They show that the problem is FPT when parameterized by the number of vertices used in an optimal solution, and use this to obtain FPT algorithms for various structural parameterizations involving the parameter ℓ, such as cluster vertex deletion number, vertex integrity, and tree-depth.

For structural parameterizations where the problem is known to be W[1]-hard, such as tree-depth and vertex integrity, the authors develop FPT approximation schemes that can efficiently compute a (1-ε)-approximate solution.

The authors prove that under the Parameterized Inapproximability Hypothesis, there is no FPT approximation scheme for the parameterization by pathwidth, even allowing running times of the form f(pw, ε)ng(ε). This precisely determines the parameter border where the problem transitions from "hard but approximable" to "inapproximable".

The authors strengthen existing lower bounds, showing that MaxNDP is XNLP-complete when parameterized by pathwidth, and improving the ETH-based lower bound for tree-depth from no(√td) to the optimal no(td).

Overall, the results provide a comprehensive understanding of the parameterized complexity and approximability of the Maximum Node-Disjoint Paths problem, highlighting the key role of the parameter ℓ in determining the tractability of the problem.

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Principais Insights Extraídos De

Parameterized Maximum Node-Disjoint Paths

by Michael Lamp... às arxiv.org 04-24-2024

https://arxiv.org/pdf/2404.14849.pdf

Perguntas Mais Profundas

Can the FPT approximation schemes developed for the cluster vertex deletion number, vertex integrity, and tree-depth parameterizations be extended to other structural parameters, such as feedback vertex set number or treewidth

The FPT approximation schemes developed for the cluster vertex deletion number, vertex integrity, and tree-depth parameterizations may not be directly extendable to other structural parameters like feedback vertex set number or treewidth. The feasibility of extending these schemes depends on the specific characteristics and relationships between the parameters in the context of the problem. While the techniques and insights gained from studying the existing parameterizations can provide a foundation for exploring new parameters, the applicability of the developed schemes to different parameters would require a thorough analysis of the problem's structure and the interplay between the parameters involved.

Is the Maximum Node-Disjoint Paths problem FPT when parameterized by the cluster vertex deletion number of the input graph

The Maximum Node-Disjoint Paths problem is not known to be FPT when parameterized solely by the cluster vertex deletion number of the input graph. The existing research and results in the provided context do not indicate a direct FPT algorithm for this parameterization. Further investigation and analysis would be needed to determine the tractability of the problem under the specific parameterization by the cluster vertex deletion number.

What other natural optimization problems in graph theory or combinatorial optimization could benefit from a similar comprehensive study of their parameterized complexity and approximability

Other natural optimization problems in graph theory or combinatorial optimization that could benefit from a similar comprehensive study of their parameterized complexity and approximability include problems like Maximum Independent Set, Minimum Vertex Cover, and Maximum Cut. These problems are fundamental in graph theory and have practical applications in various fields. By exploring their parameterized complexity and approximability, researchers can gain insights into the inherent computational complexity of these problems and develop efficient algorithms for solving them under different parameterizations. This comprehensive study can lead to a deeper understanding of the structural properties of these problems and provide valuable insights into their algorithmic behavior.