Given a finite family of squares in the plane, the packing problem asks for the maximum number $\nu$ of pairwise disjoint squares among them, while the hitting problem for the minimum number $\tau$ of points hitting all of them. Clearly, $\tau \ge \nu$. Both problems are known to be NP-hard, even for families of axis-parallel unit squares. The main results of this work provide the first non-trivial bounds for the $\tau / \nu$ ratio for not necessarily axis-parallel squares. We establish an upper bound of $6$ for unit squares and $10$ for squares of varying sizes. The worst ratios we can provide with examples are $3$ and $4$, respectively. For comparison, in the axis-parallel case, the supremum of the considered ratio is in the interval $[\frac{3}{2},2]$ for unit squares and $[\frac{3}{2},4]$ for squares of varying sizes. The methods we introduced for the $\tau/\nu$ ratio can also be used to relate the chromatic number $\chi$ and clique number $\omega$ of squares by bounding the $\chi/\omega$ ratio by $6$ for unit squares and $9$ for squares of varying sizes. The $\tau / \nu$ and $\chi/\omega$ ratios have already been bounded before by a constant for "fat" objects, the fattest and simplest of which are disks and squares. However, while disks have received significant attention, specific bounds for squares have remained essentially unexplored. This work intends to fill this gap.

翻译：给定平面上的一个有限正方形族，覆盖问题要求找出其中两两不交正方形的最大数量$\nu$，而命中问题则要求找出击中所有正方形所需的最少点数$\tau$。显然有$\tau \ge \nu$。这两个问题均被证明是NP难的，即使对于轴平行单位正方形族也是如此。本研究的主要成果首次为不限定轴平行的正方形提供了$\tau / \nu$比值的非平凡界。我们证明了单位正方形该比值的上界为$6$，变尺寸正方形该比值的上界为$10$。通过构造实例可得对应比值的最差下界分别为$3$和$4$。作为对比，在轴平行情形下，单位正方形的该比值上确界位于区间$[\frac{3}{2},2]$内，变尺寸正方形则位于$[\frac{3}{2},4]$内。我们为$\tau/\nu$比值建立的研究方法同样可用于关联正方形的色数$\chi$与团数$\omega$，证明单位正方形的$\chi/\omega$比值以$6$为界，变尺寸正方形以$9$为界。对于"胖"几何体（其中最典型且结构最简单的代表是圆盘和正方形），$\tau / \nu$与$\chi/\omega$比值的常数界已有前人研究。然而，尽管圆盘的相关问题已获得广泛关注，针对正方形的具体界值研究却基本处于空白状态。本工作旨在填补这一空白。