On central placements of new vertices in a planar point set

The vertices of an edge-weighted clique shall be placed in the plane so as to minimize the sum of all weighted distances, called the spread. Driven by practical applications in factory layout planning, we consider this problem under several constraints. First we show, in the Manhattan metric, the NP-completeness of the version where some vertices are already placed, and some minimum distance is prescribed between any two vertices. However, we can optimally append one new vertex to n placed vertices in $O\left(n^2\right)$ time. For the problem without minimum distance requirements but with many unplaced vertices, we give some structural properties of optimal solutions.