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

Abstract

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 and the best known lower bound shows only strong NP-hardness. In this paper we close this gap, assuming that we are allowed to shrink the polygons a little bit, by a factor 1 − δ for an arbitrarily small constant δ > 0, while the compared optimal solution cannot do this (resource augmentation). In this setting, we improve the approximation ratio of n to (1 + ) which matches the above lower bound that still holds if we can shrink the polygons.