Covering Polygons by Min-Area Convex Polygons

TL;DR

This paper introduces a near-linear time algorithm for covering disjoint polygons with a minimal total area of convex polygons, ensuring a unique, optimal convex cover through a convexification process.

Contribution

It presents a novel near-linear time algorithm for computing the minimum-area convex polygon cover of disjoint polygons, with a proof of uniqueness of the solution.

Findings

The convexification process yields a unique minimal-area convex cover.

The algorithm efficiently computes the convex cover in near linear time.

The method extends to covering segments with minimal-area convex polygons.

Abstract

Given a set of disjoint simple polygons $\sigma_1, \ldots, \sigma_n$, of total complexity $N$, consider a convexification process that repeatedly replaces a polygon by its convex hull, and any two (by now convex) polygons that intersect by their common convex hull. This process continues until no pair of polygons intersect. We show that this process has a unique output, which is a cover of the input polygons by a set of disjoint convex polygons, of total minimum area. Furthermore, we present a near linear time algorithm for computing this partition. The more general problem of covering a set of $N$ segments (not necessarily disjoint) by min-area disjoint convex polygons can also be computed in near linear time. A similar result is already known, see the work by Barba et al. [BBB+13].