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

Thumbnail PDF


  • Filesize: 0.61 MB
  • 13 pages

Document Identifiers

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

Cite AsGet BibTex

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


  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    PDF Downloads


  1. Boris Aronov, Esther Ezra, and Micha Sharir. Small-Size ε-Nets for Axis-Parallel Rectangles and Boxes. SIAM J. Comput., 39(7):3248-3282, 2010. URL:
  2. Nikhil Bansal and Kirk Pruhs. The Geometry of Scheduling. SIAM J. Computing, 43(5):1684-1698, 2014. URL:
  3. Hervé Brönnimann and Michael T. Goodrich. Almost Optimal Set Covers in Finite VC-Dimension. Discrete Comput. Geom., 14(4):463-479, 1995. URL:
  4. Timothy M. Chan and Elyot Grant. Exact Algorithms and APX-hardness Results for Geometric Packing and Covering Problems. Comput. Geom. Theory Appl., 47(2):112-124, 2014. URL:
  5. Timothy M. Chan, Elyot Grant, Jochen Könemann, and Malcolm Sharpe. Weighted capacitated, priority, and geometric set cover via improved quasi-uniform sampling. In Proc. 23th Annu. ACM-SIAM Symp. Discrete Algorithms (SODA'12), pages 1576-1585, 2012. URL:
  6. Aparna Das, Krzysztof Fleszar, Stephen G. Kobourov, Joachim Spoerhase, Sankar Veeramoni, and Alexander Wolff. Approximating the Generalized Minimum Manhattan Network Problem. Algorithmica, 80(4):1170-1190, 2018. URL:
  7. Irit Dinur and David Steurer. Analytical approach to parallel repetition. In Proc. Symp. Theory Comput. (STOC'14), pages 624-633, 2014. URL:
  8. Guy Even, Retsef Levi, Dror Rawitz, Baruch Schieber, Shimon Shahar, and Maxim Sviridenko. Algorithms for capacitated rectangle stabbing and lot sizing with joint set-up costs. ACM Trans. Algorithms, 4(3), 2008. URL:
  9. Uriel Feige. A Threshold of ln n for Approximating Set Cover. J. ACM, 45(4):634-652, 1998. URL:
  10. Gerd Finke, Vincent Jost, Maurice Queyranne, and András Sebö. Batch processing with interval graph compatibilities between tasks. Discrete Appl. Math., 156(5):556-568, 2008. URL:
  11. Michael R. Garey, David S. Johnson, and Larry Stockmeyer. Some simplified NP-complete problems. In Proc. 6th Annu. ACM Symp. Theory Comput. (STOC'74), pages 47-63, 1974. URL:
  12. Daya Ram Gaur, Toshihide Ibaraki, and Ramesh Krishnamurti. Constant Ratio Approximation Algorithms for the Rectangle Stabbing Problem and the Rectilinear Partitioning Problem. J. Algorithms, 43(1):138-152, 2002. URL:
  13. Panos Giannopoulos, Christian Knauer, Günter Rote, and Daniel Werner. Fixed-parameter tractability and lower bounds for stabbing problems. Comput. Geom. Theory Appl., 46(7):839-860, 2013. URL:
  14. Sofia Kovaleva and Frits C. R. Spieksma. Approximation Algorithms for Rectangle Stabbing and Interval Stabbing Problems. SIAM J. Discrete Math., 20(3):748-768, 2006. URL:
  15. Ching-Chi Lin, Hsueh-I Lu, and I-Fan Sun. Improved Compact Visibility Representation of Planar Graph via Schnyder’s Realizer. SIAM J. Discrete Math., 18(1):19-29, 2004. URL:
  16. Nabil H. Mustafa, Rajiv Raman, and Saurabh Ray. Quasi-Polynomial Time Approximation Scheme for Weighted Geometric Set Cover on Pseudodisks and Halfspaces. SIAM J. Computing, 44(6):1650-1669, 2015. URL:
  17. Nabil H. Mustafa and Saurabh Ray. Improved Results on Geometric Hitting Set Problems. Discrete Comput. Geom., 44(4):883-895, 2010. URL:
  18. Kasturi Varadarajan. Weighted geometric set cover via quasi-uniform sampling. In Proc. 42nd ACM Symp. Theory Comput. (STOC'10), pages 641-648, 2010. URL: