Independent Set of Convex Polygons: From $$n^{\epsilon }$$ to $$1+\epsilon $$1+ϵ via Shrinking

In the Independent Set of Convex Polygons problem we are given a set of weighted convex polygons in the plane and we want to compute a maximum weight subset of non-overlapping polygons. This is a very natural and well-studied problem with applications in many different areas. Unfortunately, there is a very large gap between the known upper and lower bounds for this problem. The best polynomial time algorithm we know has an approximation ratio of \(n^{\epsilon }\) and the best known lower bound shows only strong \(\mathsf {NP}\)-hardness.