Covering Simple Orthogonal Polygons with Rectangles

TL;DR

This paper demonstrates that a simple local search algorithm provides a PTAS for covering the boundary of simple orthogonal polygons with rectangles, but not for covering their interior, highlighting the problem's complexity.

Contribution

It proves the effectiveness of local search for the Boundary Cover problem on simple polygons and shows limitations for the Interior Cover problem due to complex support graph structures.

Findings

Local search yields a PTAS for Boundary Cover on simple polygons.

Support graphs for Interior Cover can have arbitrarily large bicliques, preventing PTAS.

Large locality gap exists for the Maximum Antirectangle problem.

Abstract

We study the problem of Covering Orthogonal Polygons with Rectangles. For polynomial-time algorithms, the best-known approximation factor is $O(\sqrt{\log n})$ when the input polygon may have holes [Kumar and Ramesh, STOC '99, SICOMP '03], and there is a $2$-factor approximation algorithm known when the polygon is hole-free [Franzblau, SIDMA '89]. Arguably, an easier problem is the Boundary Cover problem where we are interested in covering only the boundary of the polygon in contrast to the original problem where we are interested in covering the interior of the polygon, hence it is also referred as the Interior Cover problem. For the Boundary Cover problem, a $4$-factor approximation algorithm is known to exist and it is APX-hard when the polygon has holes [Berman and DasGupta, Algorithmica '94]. In this work, we investigate how effective is local search algorithm for the above covering problems on simple polygons. We prove that a simple local search algorithm yields a PTAS for the Boundary Cover problem when the polygon is simple. Our proof relies on the existence of planar supports on appropriate hypergraphs defined on the Boundary Cover problem instance. On the other hand, we construct instances where support graphs for the Interior Cover problem have arbitrarily large bicliques, thus implying that the same local search technique cannot yield a PTAS for this problem. We also show large locality gap for its dual problem, namely the Maximum Antirectangle problem.