Parameterized Approximation of Rectangle Stabbing

Abstract

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 $ε> 0$.

Figures (5)

• Figure 1: The set of horizontal lines $\mathcal{H}_1$ and vertical lines $\mathcal{V}_0$ computed in Lemma \ref{['lem-sweep']}. On the left, sweeping upwards, we repeatedly place a horizontal line just before the number of vertically disjoint rectangles would exceed $k$. Here $k = 3$. On the right, the remaining rectangles (highlighted) after placing $\mathcal{H}_1$ are shown in blue, and stabbed by $\mathcal{V}_0$.
• Figure 2: The vertical lines $\mathcal{V}_1$ and strips $\Gamma_v$ guaranteed by Lemma \ref{['lem-vstrip']}. On the left, the vertical dashed lines represent the unknown lines of OPT, $\mathcal{V}^*$, while the solid green are those of $\mathcal{V}_0$. On the right, the strips of $\Gamma_v$ are highlighted green. The vertical lines of $\mathcal{V}_1$ in purple include the boundaries of all heavy strips, and separate the strips of $\Gamma_v$.
• Figure 3: The subset of rectangles $\mathcal{K}$ and horizontal lines $\mathcal{H}_0$ computed in Lemma \ref{['lem-redundant']}. On the left, for each vertical green strip of $\Gamma_v$, we remove the widest rectangle (red) that is part of a large set of horizontally disjoint rectangles (yellow) touching the strip boundary, since it must be stabbed vertically. The surviving rectangles make up $\mathcal{K}$. On the right, the set of purple lines $\mathcal{H}_0$ stab all remaining unstabbed rectangles.
• Figure 4: The horizontal lines $\mathcal{H}_1'$ and strips $\Gamma_h$ guaranteed by Lemma \ref{['lem-hstrip']}. On the left, the horizontal dashed lines represent the unknown lines of OPT, $\mathcal{H}^*$, while the solid brown lines are those of $\mathcal{H}_0$ and the solid green lines are those of $\mathcal{H}_1$. On the right, the strips of $\Gamma_h$ are highlighted yellow. The horizontal lines of $\mathcal{H}_1'$ in purple include the boundaries of all heavy strips and arbitrary lines between two consecutive light strips. They separate the strips of $\Gamma_h$. All the rectangles are now stabbed by either a strip or a line.
• Figure 5: Example snippet of the reduction construction from Multicolored Clique to Rectangle Stabbing. The dark purple rectangles (line segments) with one negative coordinate are in ${\cal R}_F$. The dark blue rectangles intersecting $\zeta_v^{2i-2} \cap \zeta_h^{2j-2}$ and $\zeta_v^{2j-2} \cap \zeta_h^{2i-2}$ are in ${\cal R}_A$. The pink and cyan rectangles arranged like staircases are in ${\cal R}_E$. Notice in this example, $r = 7$ and the bottom left endpoints of the rectangles intersecting $\zeta_v^{2i-2} \cap \zeta_h^{2j-2}$ have an offset of $(1, 4)$ and $(5, 5)$ from $(2ir+1, 2jr+1)$. These represent the non-edges $(v^i_1, v^j_4),\ (v^i_5, v^j_5) \not\in E(G)$ for this example graph $G$.

Theorems & Definitions (25)

• Theorem 1.1
• Theorem 1.2
• Lemma 3.1
• proof
• Lemma 3.2
• proof
• Lemma 3.3
• proof
• Lemma 3.4
• proof