  We consider the problem of assigning radii to a given set of points in the
plane, such that the resulting set of circles is connected, and the sum of
radii is minimized. We show that the problem is polynomially solvable if a
connectivity tree is given. If the connectivity tree is unknown, the problem is
NP-hard if there are upper bounds on the radii and open otherwise. We give
approximation guarantees for a variety of polynomial-time algorithms, describe
upper and lower bounds (which are matching in some of the cases), provide
polynomial-time approximation schemes, and conclude with experimental results
and open problems.
