On the exact complexity of polyomino packing

Hans L. Bodlaender; Tom C. van der Zanden

doi:10.1016/j.tcs.2020.05.025

On the exact complexity of polyomino packing

Hans L. Bodlaender, Tom C. van der Zanden^*

^*Corresponding author for this work

Research output: Contribution to journal › Article › Academic › peer-review

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Abstract

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. (C) 2020 Elsevier B.V. All rights reserved.

Original language	English
Pages (from-to)	13-20
Number of pages	8
Journal	Theoretical Computer Science
Volume	839
DOIs	https://doi.org/10.1016/j.tcs.2020.05.025
Publication status	Published - 2 Nov 2020

Keywords

Polyomino packing
Exact complexity
Exponential time hypothesis

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Cite this

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title = "On the exact complexity of polyomino packing",

abstract = "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. (C) 2020 Elsevier B.V. All rights reserved.",

keywords = "Polyomino packing, Exact complexity, Exponential time hypothesis",

author = "Bodlaender, {Hans L.} and {van der Zanden}, {Tom C.}",

note = "Funding Information: This work was supported by the NETWORKS project, funded by the Netherlands Organization for Scientific Research NWO [project no. 024.002.003]. Publisher Copyright: {\textcopyright} 2020 Elsevier B.V.",

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doi = "10.1016/j.tcs.2020.05.025",

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T1 - On the exact complexity of polyomino packing

AU - Bodlaender, Hans L.

AU - van der Zanden, Tom C.

N1 - Funding Information: This work was supported by the NETWORKS project, funded by the Netherlands Organization for Scientific Research NWO [project no. 024.002.003]. Publisher Copyright: © 2020 Elsevier B.V.

PY - 2020/11/2

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N2 - 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. (C) 2020 Elsevier B.V. All rights reserved.

AB - 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. (C) 2020 Elsevier B.V. All rights reserved.

KW - Polyomino packing

KW - Exact complexity

KW - Exponential time hypothesis

U2 - 10.1016/j.tcs.2020.05.025

DO - 10.1016/j.tcs.2020.05.025

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