Parameterized Approximation of Rectangle Stabbing

TL;DR

This paper introduces a parameterized approximation algorithm for the Rectangle Stabbing problem that improves upon the previous 2-approximation ratio, providing solutions within 1.75 times the optimal for fixed parameters.

Contribution

First parameterized approximation algorithm with ratio better than 2 for Rectangle Stabbing, with bounds on the approximation ratio and computational complexity.

Findings

Algorithm runs in time $k^{O(k)}(|{ m{f L}}||{ m{f R}}|)^{O(1)}$

Produces solutions with at most $rac{7k}{4}$ lines

No $(rac{5}{4}- ext{epsilon})$-approximation exists under FPT ≠ W[1]

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 $(\frac{5}{4}-\epsilon)$-approximation algorithm running in $f(k)(|{\cal L}||{\cal R}|)^{O(1)}$ time for any function $f$ and $\epsilon > 0$.