Parameterized Approximation of Rectangle Stabbing

2026-04-05

AI Summary

This study addresses the parameterized approximation of the axis-aligned rectangle hitting problem: given a set of axis-aligned rectangles and a set of axis-aligned lines, the goal is to select the smallest subset of lines that intersects all rectangles. The paper presents the first parameterized algorithm achieving an approximation ratio better than the classical factor of 2, specifically attaining a 7/4-approximation in time $k^{O(k)}(|L||R|)^{O(1)}$, where $k$ denotes the size of an optimal solution. Furthermore, it establishes that, unless FPT = W[1], no fixed-parameter tractable algorithm can achieve an approximation ratio better than $5/4 - \varepsilon$ for any $\varepsilon > 0$. By integrating techniques from parameterized algorithm design, combinatorial optimization, and W[1]-hardness reductions, this work achieves a significant advance in balancing approximation guarantees with computational complexity.

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 $ε> 0$.

Problem

• Rectangle Stabbing
• parameterized approximation
• axis-parallel rectangles
• line cover
• computational geometry

Innovation

• parameterized approximation
• Rectangle Stabbing
• approximation ratio
• FPT
• computational geometry