Maximum Area Axis-Aligned Square Packings

Authors Hugo A. Akitaya , Matthew D. Jones, David Stalfa , Csaba D. Tóth

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Author Details

Hugo A. Akitaya
  • Tufts University, Medford, MA, USA
Matthew D. Jones
  • Tufts University, Medford, MA, USA
David Stalfa
  • Northeastern University, Boston, MA, USA
Csaba D. Tóth
  • California State University Northridge, Los Angeles, CA, USA

Cite AsGet BibTex

Hugo A. Akitaya, Matthew D. Jones, David Stalfa, and Csaba D. Tóth. Maximum Area Axis-Aligned Square Packings. In 43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 117, pp. 77:1-77:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)


Given a point set S={s_1,... , s_n} in the unit square U=[0,1]^2, an anchored square packing is a set of n interior-disjoint empty squares in U such that s_i is a corner of the ith square. The reach R(S) of S is the set of points that may be covered by such a packing, that is, the union of all empty squares anchored at points in S. It is shown that area(R(S))>= 1/2 for every finite set S subset U, and this bound is the best possible. The region R(S) can be computed in O(n log n) time. Finally, we prove that finding a maximum area anchored square packing is NP-complete. This is the first hardness proof for a geometric packing problem where the size of geometric objects in the packing is unrestricted.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Combinatorial optimization
  • Theory of computation → Computational geometry
  • square packing
  • geometric optimization


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  1. Kevin Balas, Adrian Dumitrescu, and Csaba D. Tóth. Anchored rectangle and square packings. Discrete Optimization, 26:131-162, 2017. URL:
  2. Kevin Balas and Csaba D. Tóth. On the number of anchored rectangle packings for a planar point set. Theor. Comput. Sci., 654:143-154, 2016. URL:
  3. Jon L. Bentley. Solutions to Klee’s rectangle problems. unpublished manuscript, 1977. Google Scholar
  4. Timothy M. Chan. Klee’s measure problem made easy. In Proc. 54th Annual IEEE Symposium on Foundations of Computer Science, pages 410-419, 2013. URL:
  5. Mark de Berg and Amirali Khosravi. Optimal binary space partitions for segments in the plane. Int. J. Comput. Geometry Appl., 22(3):187-206, 2012. URL:
  6. Adrian Dumitrescu and Csaba D. Tóth. Packing anchored rectangles. Combinatorica, 35(1):39-61, 2015. URL:
  7. Michael Formann and Frank Wagner. A packing problem with applications to lettering of maps. In Robert L. Scot Drysdale, editor, Proc. 7th Annual Symposium on Computational Geometry, pages 281-288. ACM, 1991. URL:
  8. Claudia Iturriaga and Anna Lubiw. Elastic labels around the perimeter of a map. J. Algorithms, 47(1):14-39, 2003. URL:
  9. Joo-Won Jung and Kyung-Yong Chwa. Labeling points with given rectangles. Inf. Process. Lett., 89(3):115-121, 2004. URL:
  10. Konstantinos G. Kakoulis and Ioannis G. Tollis. Labeling algorithms. In Roberto Tamassia, editor, Handbook on Graph Drawing and Visualization., pages 489-515. Chapman and Hall/CRC, 2013. URL:
  11. Klara Kedem, Ron Livne, János Pach, and Micha Sharir. On the union of Jordan regions and collision-free translational motion amidst polygonal obstacles. Discrete & Computational Geometry, 1:59-70, 1986. URL:
  12. Donald E. Knuth and Arvind Raghunathan. The problem of compatible representatives. SIAM J. Discrete Math., 5(3):422-427, 1992. URL:
  13. Atsushi Koike, Shin-Ichi Nakano, Takao Nishizeki, Takeshi Tokuyama, and Shuhei Watanabe. Labeling points with rectangles of various shapes. Int. J. Comput. Geometry Appl., 12(6):511-528, 2002. URL:
  14. Giri Narasimhan and Michiel H. M. Smid. Geometric Spanner Networks. Cambridge University Press, 2007. Google Scholar
  15. William T. Tutte, editor. Recent Progress in Combinatorics, New York, 1969. Academic Press. Proceedings of the 3rd Waterloo Conference on Combinatorics, May, 1968. Google Scholar
  16. Marc J. van Kreveld, Tycho Strijk, and Alexander Wolff. Point labeling with sliding labels. Comput. Geom., 13(1):21-47, 1999. URL:
  17. Hakan Yildiz and Subhash Suri. Computing Klee’s measure of grounded boxes. Algorithmica, 71(2):307-329, 2015. URL:
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