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I am trying to solve the problem below. It is like k-minimum-spanning-tree and the steiner tree problem, but it is with a graph.

We have a non-negative undirected weighted graph G = (V, E).

For every pair of vertices v1 and v2 there exists an edge e12. In other words, every vertex is connected to every other vertex.

We shall create a subset of the vertices U that contains k vertices.

Our goal is to select the n vertices in U such that the sum of the edges from each vertex in U to every other vertex is minimized. In other words, we want to select the vertices in U so that the sum of all the edges from the nodes in U outwards is minimized.

1 < n < number of vertices

Am I correct that neither k-MST or the Steiner tree approximation solutions will work? If so, what is this problem called? And what are the solutions? I am fine with using heuristics or approximations to solve this problem and don't need formal proofs.