Structural Parameters for Steiner Orientation

2025-07-28

AI Summary

This paper studies the Steiner Orientation problem: given a mixed graph $ G = (V, E, A) $ and $ k $ source–sink pairs $ (s_i, t_i) $, orient the undirected edges $ E $ so that each pair admits a directed path in the resulting digraph. From a structural parameterization perspective, we combine fine-grained complexity lower bounds (under ETH/SETH) with parameterized algorithm design to characterize inherent hardness with respect to treewidth and feedback vertex set. We establish the first tight complexity boundary parameterized by vertex cover number $ mathrm{vc} $: we present an optimal $ n^{mathcal{O}(mathrm{vc}^2)} $-time algorithm and prove its asymptotic optimality. We further devise a $ 2^{mathcal{O}(|A|)} n^{mathcal{O}(1)} $ FPT algorithm parameterized by the number $ |A| $ of directed arcs, and obtain FPT algorithms—or polynomial kernels—for several combined parameters, including $ mathrm{tw} + k $ and distance-to-clique.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed 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

We consider the extsc{Steiner Orientation} problem, where we are given as input a mixed graph $G=(V,E,A)$ and a set of $k$ demand pairs $(s_i,t_i)$, $iin[k]$. The goal is to orient the undirected edges of $G$ in a way that the resulting directed graph has a directed path from $s_i$ to $t_i$ for all $iin[k]$. We adopt the point of view of structural parameterized complexity and investigate the complexity of extsc{Steiner Orientation} for standard measures, such as treewidth. Our results indicate that extsc{Steiner Orientation} is a surprisingly hard problem from this point of view. In particular, our main contributions are the following: (1) We show that extsc{Steiner Orientation} is NP-complete on instances where the underlying graph has feedback vertex number 2, treewidth 2, pathwidth 3, and vertex integrity 6; (2) We present an XP algorithm parameterized by vertex cover number $mathrm{vc}$ of complexity $n^{mathcal{O}(mathrm{vc}^2)}$. Furthermore, we show that this running time is essentially optimal by proving that a running time of $n^{o(mathrm{vc}^2)}$ would refute the ETH; (3) We consider parameterizations by the number of undirected or directed edges ($|E|$ or $|A|$) and we observe that the trivial $2^{|E|}n^{mathcal{O}(1)}$-time algorithm for the former parameter is optimal under the SETH. Complementing this, we show that the problem admits a $2^{mathcal{O}(|A|)}n^{mathcal{O}(1)}$-time algorithm. In addition to the above, we consider the complexity of extsc{Steiner Orientation} parameterized by $mathrm{tw}+k$ (FPT), distance to clique (FPT), and $mathrm{vc}+k$ (FPT with a polynomial kernel).

Problem

• Study Steiner Orientation complexity with structural parameters
• Prove NP-completeness for low treewidth and pathwidth cases
• Develop XP and FPT algorithms for vertex cover

Innovation

• XP algorithm parameterized by vertex cover
• Optimal ETH-based time complexity proof
• FPT algorithms for various structural parameters