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Given an undirected graph $G = (V, E)$ and a weight function $w:E \to\mathbb{R}$, the \textsc{Minimum Dominating Tree} problem asks to find aminimum weight sub-tree of $G$, $T = (U, F)$, such that every $v \in V\setminus U$ is adjacent to at least one vertex in $U$. The special case whenthe weight function is uniform is known as the \textsc{Minimum ConnectedDominating Set} problem.  Given an undirected graph $G = (V, E)$ with some subsets of vertices calledgroups, and a weight function $w:E \to \mathbb{R}$, the \textsc{Group SteinerTree} problem is to find a minimum weight sub-tree of $G$ which contains atleast one vertex from each group.  In this paper we show that the two problems are equivalents fromapproximability perspective. This improves upon both the best knownapproximation algorithm and the best inapproximability result for the\textsc{Minimum Dominating Tree} problem. We also consider two extrema variantsof the \textsc{Minimum Dominating Tree} problem, namely, the \textsc{MinimumDominating Star} and the \textsc{Minimum Dominating Path} problems which ask tofind a minimum dominating star and path respectively.

Hardness Results and Approximation Algorithms for the Minimum Dominating Tree Problem