In the Rectangle Stabbing problem, input is a set ${\cal R}$ of axis-parallel rectangles and a set ${\cal L}$ of axis parallel lines in the plane. The task is to find a minimum size set ${\cal L}^* \subseteq {\cal L}$ such that for every rectangle $R \in {\cal R}$ there is a line $\ell \in {\cal L}^*$ such that $\ell$ intersects $R$. Gaur et al. [Journal of Algorithms, 2002] gave a polynomial time $2$-approximation algorithm, while Dom et al. [WALCOM 2009] and Giannopolous et al. [EuroCG 2009] independently showed that, assuming FPT $\neq$ W[1], there is no algorithm with running time $f(k)(|{\cal L}||{\cal R}|)^{O(1)}$ that determines whether there exists an optimal solution with at most $k$ lines. We give the first parameterized approximation algorithm for the problem with a ratio better than $2$. In particular we give an algorithm that given ${\cal R}$, ${\cal L}$, and an integer $k$ runs in time $k^{O(k)}(|{\cal L}||{\cal R}|)^{O(1)}$ and either correctly concludes that there does not exist a solution with at most $k$ lines, or produces a solution with at most $\frac{7k}{4}$ lines. We complement our algorithm by showing that unless FPT $=$ W[1], the Rectangle Stabbing problem does not admit a $(\frac{5}{4}-ε)$-approximation algorithm running in $f(k)(|{\cal L}||{\cal R}|)^{O(1)}$ time for any function $f$ and $ε> 0$.

翻译：在矩形刺穿问题中，输入为平面上的轴平行矩形集合 ${\cal R}$ 与轴平行直线集合 ${\cal L}$。任务要求找到最小规模子集 ${\cal L}^* \subseteq {\cal L}$，使得每个矩形 $R \in {\cal R}$ 至少被一条直线 $\ell \in {\cal L}^*$ 相交。Gaur 等人 [Journal of Algorithms, 2002] 给出了多项式时间的 $2$-近似算法，而 Dom 等人 [WALCOM 2009] 与 Giannopolous 等人 [EuroCG 2009] 独立证明：若假设 FPT $\neq$ W[1]，则不存在运行时间为 $f(k)(|{\cal L}||{\cal R}|)^{O(1)}$ 的算法来判定是否存在至多 $k$ 条直线的最优解。本文给出了该问题首个比率优于 $2$ 的参数化近似算法。具体而言，我们提出一种算法：给定 ${\cal R}$、${\cal L}$ 和整数 $k$，其运行时间为 $k^{O(k)}(|{\cal L}||{\cal R}|)^{O(1)}$，并能够正确判定不存在至多 $k$ 条直线的解，或给出一个至多包含 $\frac{7k}{4}$ 条直线的解。我们通过证明以下结论对算法进行补充：除非 FPT $=$ W[1]，否则对于任意函数 $f$ 和 $\epsilon > 0$，矩形刺穿问题不存在运行时间为 $f(k)(|{\cal L}||{\cal R}|)^{O(1)}$ 的 $(\frac{5}{4}-\epsilon)$-近似算法。