The Rectilinear Steiner Forest Arborescence problem

Let $r$ be a point in the first quadrant $Q_1$ of the plane $\mathbb{R}^2$ and let $P \subset Q_1$ be a set of points such that for any $p \in P$, its $x$- and $y$-coordinate is at least as that of $r$. A rectilinear Steiner arborescence for $P$ with the root $r$ is a rectilinear Steiner tree $T$ for $P \cup \{r\}$ such that for each point $p \in P$, the length of the (unique) path in $T$ from $p$ to the root $r$ equals $({\rm x}(p)-{\rm x}(r))+({\rm y}(p))-({\rm y}(r))$, where ${\rm x}(q)$ and ${\rm y}(q)$ denote the $x$- and $y$-coordinate, respectively, of point $q \in P \cup \{r\}$. Given two point sets $P$ and $R$ lying in the first quadrant $Q_1$ and such that $(0,0) \in R$, the Rectilinear Steiner Forest Arborescence (RSFA) problem is to find the minimum-length spanning forest $F$ such that each connected component $F$ is a rectilinear Steiner arborescence rooted at some root in $R$. The RSFA problem is a natural generalization of the Rectilinear Steiner Arborescence problem, where $R=\{(0,0)\}$, and thus it is NP-hard. In this paper, we provide a simple exact exponential time algorithm for the RSFA problem, design a polynomial time approximation scheme as well as a fixed-parameter algorithm.