Given a family ${\mathcal F}$ of shapes in the plane, we study what is the lowest possible density of a point set $P$ that pierces (``intersects'', ``hits'') all translates of each shape in ${\mathcal F}$. For instance, if ${\mathcal F}$ consists of two axis-parallel rectangles the best known piercing set, i.e., one with the lowest density, is a lattice. Given a finite family ${\mathcal F}$ of axis-parallel rectangles, we present an algorithm for finding an optimal ${\mathcal F}$-piercing lattice. The algorithm runs in time polynomial in the number of rectangles and the maximum aspect ratio of the rectangles in the family. No prior algorithms for this problem were known. On the other hand, we show that for every $n \geq 3$, there exists a family of $n$ axis-parallel rectangles for which the best piercing density achieved by a lattice is separated by a positive (constant) gap from the optimal piercing density for the respective family. Finally, we show that the best lattice can be sometimes worse by $20\%$ than the optimal piercing set.

翻译：给定平面上的一个形状族 ${\\mathcal F}$，我们研究能够穿透（即“相交”或“击中”）${\\mathcal F}$ 中每个形状的所有平移版本的点集 $P$ 可能达到的最低密度。例如，若 ${\\mathcal F}$ 包含两个轴平行矩形，目前已知的最佳穿透集（即密度最低的集合）是一个格点集。针对有限个轴平行矩形构成的族 ${\\mathcal F}$，我们提出一种寻找最优 ${\\mathcal F}$-穿透格点的算法。该算法运行时间与矩形数量及族中矩形的最大长宽比呈多项式关系。此前该问题尚无已知算法。另一方面，我们证明对于任意 $n \\geq 3$，存在一个由 $n$ 个轴平行矩形构成的族，使得格点所能达到的最佳穿透密度与该族对应的最优穿透密度之间存在正（常数）差距。最后，我们通过实例说明最佳格点集的穿透效果有时可能比最优穿透集差 $20\\%。