Covering Simple Orthogonal Polygons with Rectangles

TL;DR

The paper investigates covering simple orthogonal polygons with axis-aligned rectangles, focusing on boundary versus interior coverage and the efficacy of local-search based PTAS approaches. It proves that the boundary-cover hypergraph for a simple polygon admits a planar support, enabling a Polynomial Time Approximation Scheme via the Mustafa–Ray local-search framework, and extends this to corner covers. Conversely, it demonstrates that the same local-search strategy cannot yield a PTAS for interior cover due to large bicliques in minimal supports, and shows a large locality gap for the Maximum Antirectangle problem. The results advance understanding of when PTAS via local search is feasible in geometric covering and identify clear limitations and open questions for interior and corner variants.

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.

Figures (3)

• Figure 1: On the left, the boundary of polygon $P$ is drawn with thick black lines on an integer grid. ${\mathcal{R}} = \{A,B,C,D,P,Q,R,S\}$ where $A = [1,4]\times[1,11]$, $B = [3,6]\times[0,10]$, $C = [5,8]\times[1,11]$, $D = [7,10]\times[0,10]$, $P = [0,10]\times[7,10]$, $Q = [1,11]\times[5,8]$, $R = [0,10]\times[3,6]$, $S = [1,11]\times[1,4]$. A support graph for the boundary cover instance ${\mathcal{R}}$ with respect to $P$ is drawn on the right. Let $R_c = [1,10]\times[1,10]$. The minimal support graph for the boundary cover instance ${\mathcal{R}}\cup\{R_c\}$ is a star graph with $R_c$ as the center of the star.
• Figure 2: The left pair has a corner intersection. The middle and the right have a piercing intersection; in which the right pair are also aligned to each other.
• Figure 3: $(\{R_i, R_j, R_k, R_\ell, R_c\}, \partial P)$ has a non-empty kernel $\{R_c\}$ but it is not proper. Whereas, $(\{R_i, R_j, R_c\}, \partial P)$ is proper and has the same kernel. Note that $\partial P$ is drawn with red lines.

Theorems & Definitions (14)

• Theorem 1: Planar Support for Boundary Cover
• Corollary 2
• Corollary 3
• Proposition 4: Proposition 5.2 in roughgarden2016communication
• Definition 5: Blocker of a Maximal Rectangle
• Definition 6: Support graph of a hypergraph
• Lemma 7: Local Search Lemma MR10
• Definition 8
• Definition 9: DFS-orientation
• Definition 10: Left-Right Coloring