Structural Parameters for Steiner Orientation

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).