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Parameterized Approximation of Rectangle Stabbing

Authors Huairui Chu , Ajaykrishnan E S , Daniel Lokshtanov , Anikait Mundhra , Thomas Schibler , Xiaoyang Xu , Jie Xue

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LIPIcs.ESA.2026.69.pdf
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Subject Classification

ACM Subject Classification
  • Theory of computation → Problems, reductions and completeness
  • Theory of computation → Computational geometry
  • Theory of computation → Packing and covering problems
  • Theory of computation → Fixed parameter tractability
  • Theory of computation → W hierarchy
Keywords
  • rectangle stabbing
  • parameterized algorithms
  • approximation algorithms
  • computational geometry
  • parameterized approximation complexity
  • geometric hitting set
  • lower bounds

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Abstract

In the Rectangle Stabbing problem, input is a set R of axis-parallel rectangles and a set L of axis-parallel lines in the plane. The task is to find a minimum size set L^* ⊆ L such that for every rectangle R ∈ R there is a line 𝓁 ∈ L^* such that 𝓁 intersects R. Gaur et al. [Journal of Algorithms, 2002] gave a polynomial time 2-approximation algorithm, while Dom et al. [WALCOM 2009] and Giannopoulos et al. [EuroCG 2009] independently showed that, assuming FPT ≠ W[1], there is no algorithm with running time f(k)(|L||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 R, L, and an integer k runs in time k^O(k)(|L||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 7k/4 lines. We complement our algorithm by showing that unless FPT = W[1], the Rectangle Stabbing problem does not admit a (5/4-ε)-approximation algorithm running in f(k)(|L||R|)^O(1) time for any function f and ε > 0.

Cite As Get BibTex

Huairui Chu, Ajaykrishnan E S, Daniel Lokshtanov, Anikait Mundhra, Thomas Schibler, Xiaoyang Xu, and Jie Xue. Parameterized Approximation of Rectangle Stabbing. In 34th Annual European Symposium on Algorithms (ESA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 388, pp. 69:1-69:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.ESA.2026.69

Author Details

Huairui Chu
  • Department of Computer Science, University of California Santa Barbara, CA, USA
Ajaykrishnan E S
  • Department of Computer Science, University of California Santa Barbara, CA, USA
Daniel Lokshtanov
  • Department of Computer Science, University of California Santa Barbara, CA, USA
Anikait Mundhra
  • Department of Computer Science, University of California Santa Barbara, CA, USA
Thomas Schibler
  • Department of Computer Science, University of California Santa Barbara, CA, USA
Xiaoyang Xu
  • Department of Computer Science, University of California Santa Barbara, CA, USA
Jie Xue
  • Department of Computer Science, New York University Shanghai, China

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