I was wondering about the complexity of the following covering problem. Let $B_i,\,i=1,\ldots,n$ be a set of unit disks in $\mathbb{R}^2$. The problem is to decide whether there exists $C\subset\{1,\ldots,n\}$ with $|C|\leq k$ such that $\bigcup_{i\in C} B_i = \bigcup_{i=1}^n B_i$. This could be considered as a continuous version of the geometric set cover problem.

There is a paper that provides a constant approximation algorithm (Basappa et al, "Unit disk cover problem in 2D") but no proof of NP completeness has been provided. In this paper, the problem is referred to as the "rectangular region cover" problem. Also, in one version of the paper, they claim that the problem is NP-complete and they refer to Garey & Johnson but I could not find a proof there either. Computational complexity is far to my area of expertise so I do not want to reinvent the wheel by reproducing some well-known result. Any help will be greatly appreciated.