Structural Parameters for Steiner Orientation

TL;DR

This paper investigates the computational complexity of the Steiner Orientation problem under various structural graph parameters, revealing its inherent hardness and providing algorithms with tight complexity bounds based on parameterized complexity theory.

Contribution

It establishes NP-completeness for key structural parameters, introduces an XP algorithm parameterized by vertex cover, and explores fixed-parameter tractability and optimality results for related parameters.

Findings

NP-complete on graphs with feedback vertex number 2

XP algorithm with vertex cover parameter, optimal under ETH

Optimal algorithms under SETH for edge-based parameters

Abstract

We consider the \textsc{Steiner Orientation} problem, where we are given as input a mixed graph $G=(V,E,A)$ and a set of $k$ demand pairs $(s_i,t_i)$, $i\in[k]$. 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 $i\in[k]$. We adopt the point of view of structural parameterized complexity and investigate the complexity of \textsc{Steiner Orientation} for standard measures, such as treewidth. Our results indicate that \textsc{Steiner Orientation} is a surprisingly hard problem from this point of view. In particular, our main contributions are the following: (1) We show that \textsc{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 \textsc{Steiner Orientation} parameterized by $\mathrm{tw}+k$ (FPT), distance to clique (FPT), and $\mathrm{vc}+k$ (FPT with a polynomial kernel).