Parameterized Maximum Node-Disjoint Paths (2404.14849v1)

Abstract: We revisit the Maximum Node-Disjoint Paths problem, the natural optimization version of Node-Disjoint Paths, where we are given a graph $G$, $k$ pairs of vertices $(s_i, t_i)$ and an integer $\ell$, and are asked whether there exist at least $\ell$ vertex-disjoint paths in $G$ whose endpoints are given pairs. We present several results, with an emphasis towards FPT approximation. Our main positive contribution is to show that the problem's intractability can be overcome using approximation and that for several of the structural parameters for which the problem is hard, most notably tree-depth, it admits an efficient FPT approximation scheme, returning a $(1-\varepsilon)$-approximate solution in time $f(td,\varepsilon)n^{O(1)}$. We manage to obtain these results by comprehensively mapping out the structural parameters for which the problem is FPT if $\ell$ is also a parameter, hence showing that understanding $\ell$ as a parameter is key to the problem's approximability. This, in turn, is a problem we are able to solve via a surprisingly simple color-coding algorithm, which relies on identifying an insightful problem-specific variant of the natural parameter, namely the number of vertices used in the solution. A natural question is whether the FPT approximation algorithm we devised for tree-depth can be extended to pathwidth. We resolve this negatively, showing that under the Parameterized Inapproximability Hypothesis no FPT approximation scheme for this parameter is possible, even in time $f(pw,\varepsilon)n^{g(\varepsilon)}$, thus precisely determining the parameter border where the problem transitions from hard but approximable'' toinapproximable''. Lastly, we strengthen existing lower bounds by replacing W[1]-hardness by XNLP-completeness for parameter pathwidth, and improving the $n^{o(\sqrt{td})}$ ETH-based lower bound for tree-depth to $n^{o(td)}$.

Overview of "Parameterized Maximum Node-Disjoint Paths"

The paper "Parameterized Maximum Node-Disjoint Paths," by Michael Lampis and Manolis Vasilakis, revisits the Maximum Node-Disjoint Paths (MaxNDP) problem within the contexts of parameterized complexity and approximability. The MaxNDP problem is a generalization of the classic NP-complete Node-Disjoint Paths problem where one seeks to determine if at least ℓ out of k given pairs of vertices in a graph can be connected by vertex-disjoint paths. This research provides a comprehensive analysis of the MaxNDP problem's tractability and offers insights into its complexity landscape under various structural parameterizations.

Contributions and Results

The authors advance the field of parameterized complexity by demonstrating that the apparent intractability of the MaxNDP problem, which has been proven hard for certain structural parameters such as tree-depth and pathwidth, can be navigated using approximation techniques. The major contributions of the paper are summarized as follows:

1. FPT Approximation Schemes: For hard structural parameters, notably tree-depth, the paper establishes efficient Fixed-Parameter Tractable (FPT) Approximation Schemes that yield (1 − ε)-approximate solutions, achievable in time f(td, ε)nO(1). This result shows that using the size of the optimal solution as a parameter, the MaxNDP problem can be approximated efficiently, offering substantial improvements in identifying the boundaries of parameter tractability.
2. The Border of Approximability: The research shows a sharp delineation in parameter space, where the MaxNDP transitions from 'hard but approximable' to 'inapproximable'. This is achieved by proving that no FPT approximation scheme exists for pathwidth under the Parameterized Inapproximability Hypothesis, thus addressing a critical question in the domain of parameterized algorithms.
3. XNLP-completeness and Tight Lower Bounds: The paper establishes that the MaxNDP problem, parameterized by pathwidth, is XNLP-complete, thus extending the known hardness to all W[t] classes for any integer t ≥ 1. Additionally, it refines known hardness results by improving the lower bound for tree-depth using ETH, showing nO(td) as optimal for MaxNDP.
4. Broad Parameterization Study: An array of structural parameters including cluster vertex deletion number and vertex integrity are methodically explored. The paper identifies novel FPT algorithms for these parameters, leveraging a simple yet effective color-coding strategy.

Implications and Future Directions

The results provide critical advancements in understanding the complexity boundaries of the MaxNDP problem, particularly in how intractability can be mitigated via approximation. The development of FPT approximation schemes not only enhances theoretical understanding but could also impact practical applications involving routing, network design, and resource allocation tasks where such configurations are frequent.

Future research directions may include exploring constant-factor approximation algorithms for different parameters and investigating the complexity implications under dynamic graph settings. Additionally, the potential to streamline the running times of the existing parameterized algorithms, which presently extend to double-exponential complexities, offers another avenue for research.

In conclusion, the paper's systematic dissection of MaxNDP underscores the complex interplay of graph parameters in algorithmic tractability, setting a foundation for further explorations in structural graph theory and parameterized complexity domains.