Belangrijkste concepten
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.
Samenvatting
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.