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Stabbing Rectangles by Line Segments - How Decomposition Reduces the Shallow-Cell Complexity

Authors Timothy M. Chan, Thomas C. van Dijk , Krzysztof Fleszar , Joachim Spoerhase , Alexander Wolff

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Author Details

Timothy M. Chan
  • University of Illinois at Urbana-Champaign, U.S.A.
Thomas C. van Dijk
  • Universität Würzburg, Germany
Krzysztof Fleszar
  • Max-Planck-Institut für Informatik, Saarbrücken, Germany
Joachim Spoerhase
  • Aalto University, Espoo, Finland, Universität Würzburg, Germany
Alexander Wolff
  • Universität Würzburg, Germany

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Timothy M. Chan, Thomas C. van Dijk, Krzysztof Fleszar, Joachim Spoerhase, and Alexander Wolff. Stabbing Rectangles by Line Segments - How Decomposition Reduces the Shallow-Cell Complexity. In 29th International Symposium on Algorithms and Computation (ISAAC 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 123, pp. 61:1-61:13, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)


We initiate the study of the following natural geometric optimization problem. The input is a set of axis-aligned rectangles in the plane. The objective is to find a set of horizontal line segments of minimum total length so that every rectangle is stabbed by some line segment. A line segment stabs a rectangle if it intersects its left and its right boundary. The problem, which we call Stabbing, can be motivated by a resource allocation problem and has applications in geometric network design. To the best of our knowledge, only special cases of this problem have been considered so far. Stabbing is a weighted geometric set cover problem, which we show to be NP-hard. While for general set cover the best possible approximation ratio is Theta(log n), it is an important field in geometric approximation algorithms to obtain better ratios for geometric set cover problems. Chan et al. [SODA'12] generalize earlier results by Varadarajan [STOC'10] to obtain sub-logarithmic performances for a broad class of weighted geometric set cover instances that are characterized by having low shallow-cell complexity. The shallow-cell complexity of Stabbing instances, however, can be high so that a direct application of the framework of Chan et al. gives only logarithmic bounds. We still achieve a constant-factor approximation by decomposing general instances into what we call laminar instances that have low enough complexity. Our decomposition technique yields constant-factor approximations also for the variant where rectangles can be stabbed by horizontal and vertical segments and for two further geometric set cover problems.

Subject Classification

ACM Subject Classification
  • Theory of computation → Packing and covering problems
  • Theory of computation → Computational geometry
  • Geometric optimization
  • NP-hard
  • approximation
  • shallow-cell complexity
  • line stabbing


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