# Node weighted Steiner Tree Problem where all Nodes have the same Weight

The node weighted Steiner Tree Problem as found in this compendium:

$$\textbf{Instance}: \text{Graph } G = (V, E)\text{, set of terminals } S \subseteq V \text{ and a node weight function } w:V \to \mathbb{R}^+$$
$$\textbf{Solution}: \text{A tree } T = (V_T,E_T) \text{ in } G \text{, such that } S\subseteq V_T \subseteq V,~E_T \subseteq E.$$
$$\textbf{Objective}: \text{Minimize the sum of the vertices weight}$$

Is NP hard. But what if all nodes have the same weight? So the weight function $$w$$ is just a constant function that maps to 1 for all Vertices: $$\forall v \in V:w(v) = 1$$

Is there a name for this Problem? Is it NP hard? If yes what algorithms can I use for approximation?

Please note that this problem is not equals MST, ShortestPath or TSP

Since all vertex weights are equal, minimizing $$\sum_{v \in V_T} w(v)$$ is equivalent to minimizing $$|V_T|$$, and since a tree on $$x$$ vertices has exactly $$x-1$$, edges, this is equivalent to minimizing $$|E_T|$$.
Moreover, any $$\alpha$$-approximation algorithm for the classical Steiner tree problem, where $$\alpha$$ is a constant, can also be used to provide a $$\alpha$$-approximation for your version. Let $$v$$ be a terminal and let $$\overline{w}$$ be the (constant) vertex weight. Add a new vertex $$v'$$ and the edge $$(v, v')$$. Then drop all vertex weights, and set all edge weights to $$\overline{w}$$.
You are left with an instance of Steiner tree such that any solution to this instance can be converted to a solution for your version of the problem having the same cost (by simply deleting $$(v,v')$$ from $$T_V$$), and vice-versa (by adding $$(v,v')$$ to $$T_V$$).