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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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