On the Exact Complexity of Polyomino Packing

Authors Hans L. Bodlaender, Tom C. van der Zanden

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Author Details

Hans L. Bodlaender
  • Department of Computer Science, Utrecht University, Utrecht, The Netherlands and Department of Mathematics and Computer Science, Eindhoven University of Technology, Eindhoven, The Netherlands
Tom C. van der Zanden
  • Department of Computer Science, Utrecht University, Utrecht, The Netherlands

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Hans L. Bodlaender and Tom C. van der Zanden. On the Exact Complexity of Polyomino Packing. In 9th International Conference on Fun with Algorithms (FUN 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 100, pp. 9:1-9:10, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)


We show that the problem of deciding whether a collection of polyominoes, each fitting in a 2 x O(log n) rectangle, can be packed into a 3 x n box does not admit a 2^{o(n/log{n})}-time algorithm, unless the Exponential Time Hypothesis fails. We also give an algorithm that attains this lower bound, solving any instance of polyomino packing with total area n in 2^{O(n/log{n})} time. This establishes a tight bound on the complexity of Polyomino Packing, even in a very restricted case. In contrast, for a 2 x n box, we show that the problem can be solved in strongly subexponential time.

Subject Classification

ACM Subject Classification
  • Theory of computation → Problems, reductions and completeness
  • Mathematics of computing → Combinatorial algorithms
  • polyomino packing
  • exact complexity
  • exponential time hypothesis


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  1. Hans L. Bodlaender, Jesper Nederlof, and Tom C. van der Zanden. Subexponential time algorithms for embedding h-minor free graphs. In Ioannis Chatzigiannakis, Michael Mitzenmacher, Yuval Rabani, and Davide Sangiorgi, editors, 43rd International Colloquium on Automata, Languages, and Programming, ICALP 2016, July 11-15, 2016, Rome, Italy, volume 55 of LIPIcs, pages 9:1-9:14. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2016. URL: http://dx.doi.org/10.4230/LIPIcs.ICALP.2016.9.
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