The Rectilinear Steiner Forest Arborescence problem
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Let r be a point in the first quadrant Q_1 of the plane ℝ^2 and let P ⊂ Q_1 be a set of points such that for any p ∈ 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 ∪{r} such that for each point p ∈ P, the length of the (unique) path in T from p to the root r equals ( x(p)- x(r))+( y(p))-( y(r)), where x(q) and y(q) denote the x- and y-coordinate, respectively, of point q ∈ P ∪{r}. Given two point sets P and R lying in the first quadrant Q_1 and such that (0,0) ∈ 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.
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