It is known that given a polygon $P$ with holes it is NP-hard to find a decomposition of $P$ into convex polygons, s.t. their number is minimized (even if Steiner points are allowed).
I am wondering if anything is known about the following problem:
Given a convex polygon $P$ which contains a collection $\mathcal{Q}$ of convex polygons in its interior, find a decomposition of $P$ into convex polygons such that no such polygon contains more than one polygon from $\mathcal{Q}$ and the number of convex polygons is minimized.
In the example below, $P$ is the black polygon, $\mathcal{Q}$ are the four blue polygons, and the red linesegments induce a convex partitioning of $P$ where each face contains at most one polygon from $\mathcal{Q}$ .
