Parameterized Maximum Node-Disjoint Paths

2024-04-23

arXiv.org

AI Summary

This paper studies the parameterized approximation of the Maximum Node-Disjoint Paths problem: given a graph $G$, $k$ terminal pairs $(s_i,t_i)$, and an integer $ell$, decide whether there exist at least $ell$ pairwise node-disjoint $s_i$–$t_i$ paths. Regarding classical structural parameters—tree-depth ($td$) and pathwidth ($pw$)—we establish three main results. First, we present the first $(1-varepsilon)$-FPT approximation algorithm parameterized by tree-depth, running in time $f(td,varepsilon)cdot n^{O(1)}$. Second, we refute the existence of any FPT approximation algorithm parameterized by pathwidth by proving XNLP-completeness. Third, we strengthen ETH-based lower bounds for tree-depth, precisely characterizing the boundary between approximability and inapproximability. Our techniques integrate color-coding, structural graph decomposition, and fine-grained complexity reductions.

Technology Category

• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Search and Optimization: Combinatorial Optimization
• Machine Learning: Graph-based Machine Learning

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Web Mining and Content Analysis: Bridging structured and unstructured data

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'' to ``inapproximable''. 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 $n^{o(td)}$.

Problem

• Develop FPT approximation schemes for Maximum Node-Disjoint Paths problem
• Determine structural parameters enabling efficient parameterized approximation algorithms
• Establish precise boundaries between approximable and inapproximable parameter regimes

Innovation

• FPT approximation scheme for tree-depth
• Color-coding algorithm for solution identification
• Pathwidth inapproximability under PIH hypothesis