直线细分的覆盖与填充
Covering and Packing of Rectilinear Subdivision
摘要：
研究了平面上有界区域的一类几何覆盖与填充问题。我们给出了一组轴平行的线段，诱导一组有界（直线）面的平面细分。我们对以下问题感兴趣。(P1)刺入-细分：通过在平面中选择最少数量的点来刺入所有有界面。(P2)独立-细分：选择成对不相交有界面的最大大小集合。(P3)主细分：选择最小尺寸的面集合，使得任何其他面具有非空交点（即，共享边或顶点）。我们证明了这些问题是NP困难的。我们甚至证明，这些问题是NP-困难的，当我们只集中在矩形面的细分。此外，我们提供了恒定因子近似算法的刺细分问题。
英文摘要：
We study a class of geometric covering and packing problems for bounded regions on the plane. We are given a set of axis-parallel line segments that induces a planar subdivision with a set of bounded (rectilinear) faces. We are interested in the following problems. (P1) Stabbing-Subdivision: Stab all bounded faces by selecting a minimum number of points in the plane. (P2) Independent-Subdivision: Select a maximum size collection of pairwise non-intersecting bounded faces. (P3) Dominating-Subdivision: Select a minimum size collection of faces such that any other face has a non-empty intersection (i.e., sharing an edge or a vertex) with some selected faces. We show that these problems are NP-hard. We even prove that these problems are NP-hard when we concentrate only on the rectangular faces of the subdivision. Further, we provide constant factor approximation algorithms for the Stabbing-Subdivision problem.
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