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.

翻译：我们研究用矩形覆盖正交多边形的问题。对于多项式时间算法，当输入多边形可能包含孔洞时，已知最佳逼近因子为$O(\sqrt{\log n})$[Kumar and Ramesh, STOC '99, SICOMP '03]；而当多边形无孔洞时，存在已知的$2$因子逼近算法[Franzblau, SIDMA '89]。相对而言，一个更简单的问题是边界覆盖问题，该问题仅关注覆盖多边形的边界，而原始问题则关注覆盖多边形内部，故亦称为内部覆盖问题。对于边界覆盖问题，已知存在$4$因子逼近算法，且当多边形有孔洞时该问题是APX难的[Berman and DasGupta, Algorithmica '94]。本文中，我们探究局部搜索算法对于上述简单多边形覆盖问题的有效性。我们证明，当多边形为简单多边形时，一个简单的局部搜索算法可为边界覆盖问题产生一个PTAS。我们的证明依赖于在边界覆盖问题实例上定义的适当超图中平面支撑的存在性。另一方面，我们构造了内部覆盖问题的支撑图包含任意大二分团的实例，从而表明相同的局部搜索技术无法为该问题产生PTAS。我们还展示了其对偶问题——最大反矩形问题——具有较大的局部间隙。