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We consider the problem of assigning radii to a given set of points in the plane, such that the resulting set of disks is connected, and the sum of radii is minimized. We prove that the problem is NP-hard in planar weighted graphs if there are upper bounds on the radii and sketch a similar proof for planar point sets. For the case when there are no upper bounds on the radii, the complexity is open;...
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...
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