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

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Abstract

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.

Cite As Get 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) https://doi.org/10.4230/LIPIcs.MFCS.2018.77

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

Funding

Research supported in part by the NSF awards CCF-1422311 and CCF-1423615. The first author was supported by the Science Without Borders program.

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