Inflating Balls is NP-Hard

A collection C of balls in R is δ-inflatable if it is isometric to the intersection U ∩ E of some d-dimensional affine subspace E with a collection U of (d + δ)-dimensional balls that are disjoint and have equal radius. We give a quadratic-time algorithm to recognize 1-inflatable collections of balls in any fixed dimension, and show that recognizing δ-inflatable collections of d-dimensional balls is NP-hard for δ ≥ 2 and d ≥ 3 if the balls’ centers and radii are given by numbers of the form a + b √ c+ d √ e, where a, . . . , e are integers.