Hitting or avoiding balls in Euclidean space

The authors investigate the algorithmic complexity of several geometric problems of the following type: given a ‘feasible’ box and a collection of balls in Euclidean space, find a feasible point which is covered by as few or, respectively, by as many balls as possible. They establish that all these problems are NP-hard in their most general version. The authors derive tight lower and upper bounds on the complexity of their one-dimensional versions. Finally, they show that all these problems can be solved in polynomial time when the dimension of the space is fixed.