discrete geometry – On Cutting Disks from Planar Regions

Question: Given a planar region R of unit area and an integer n, to cut n circular disks (their sizes need not be equal) such that the highest fraction of R is taken off.

A simple greedy algorithm would be to repeatedly cut the biggest disk that can be cut at each stage from what remains of R. I don’t know of any R and n where this algorithm clearly fails to give an optimal result. Ideally, one is looking for a convex R and some n such that the greedy algo clearly fails.

Remarks: If from a given R, instead of disks, n triangles or n squares are to be cut such that max fraction of R is to be removed, ‘greedy’ clearly fails in general. For simple examples, see http://nandacumar.blogspot.com/2020/12/breaking-pieces-from-polygons.html

One can ask whether the problem of cutting say n triangles off from a unit area m-gon is NP complete.