On the complexity of two circle strongly connecting problems

Given n demand points in the plane, the circle strongly connecting problem (CSCP) is to locate n circles in the plane, each with its center in a different demand point, and determine the radius of each circle such that the corresponding digraph G=( V, E), in which a vertex ν₁ in V stands for the point p_i, and a directed edge ⟨ν_i, ν_j⟩ in E, if and only if p_j located within the circle of p _i, is strongly connected, and the sum of the radii of these n circles is minimal. The constrained circle strongly connecting problem is similar to the CSCP except that the points are given in the plane with a set of obstacles and a directed edge ⟨ν_i, ν_j⟩ in E, if and only if p_j is located within the circle of p_i and no obstacles exist between them. It is proven that both these geometric problems are NP-hard. An O( n log n) approximation algorithm that can produce a solution no greater than twice an optimal one is also proposed