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Dissection with the Fewest Pieces is Hard, Even to Approximate

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Discrete and Computational Geometry and Graphs (JCDCGG 2015)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9943))

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Abstract

We prove that it is NP-hard to dissect one simple orthogonal polygon into another using a given number of pieces, as is approximating the fewest pieces to within a factor of \(1+1/1080-\varepsilon \).

P. Manurangsi—Part of this work was completed while the author was at Massachusetts Institute of Technology and Dropbox, Inc.

A. Yodpinyanee—Research supported by NSF grant CCF-1420692.

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Notes

  1. 1.

    Given a 4-Partition instance \(A = \{a_1, \dots , a_n\}\), we can create a 5-Partition instance by setting \(A' = \{na_1, \dots , na_n, 1, \dots , 1\}\) where the number of 1s is n / 4.

  2. 2.

    The best \(\varepsilon _{\mathrm {MPD}}\) we can achieve is \(1/1080 - \varepsilon \) for any \(\varepsilon \in (0, 1/1080)\).

  3. 3.

    Because \(k = n\), \(a_1\) will remain attached to the bar, forcing it to be the first element rectangle placed in the first partition rectangle. Because the order of and within partitions does not matter, this constraint does not affect the 5-Partition simulation.

  4. 4.

    As stated earlier, the result from [10] is for 4-Partition, but 4-Partition is easily reduced to 5-Partition; see Sect. 2.

  5. 5.

    The best \(\alpha _{\mathrm {M5P}}\) we can achieve here is 216/215.

  6. 6.

    Our reduction uses rational lengths, but the polygons can be scaled up to use integer lengths while still being of polynomial size.

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Acknowledgment

We thank Greg Frederickson for helpful discussions.

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Authors and Affiliations

  1. Computer Science and AI Laboratory, Massachusetts Institute of Technology, 32 Vassar St., Cambridge, Massachusetts, 02139, USA

    Jeffrey Bosboom, Erik D. Demaine, Martin L. Demaine, Jayson Lynch, Mikhail Rudoy & Anak Yodpinyanee

  2. University of California, Berkeley, California, 94720, USA

    Pasin Manurangsi

Authors
  1. Jeffrey Bosboom
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  2. Erik D. Demaine
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  3. Martin L. Demaine
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  4. Jayson Lynch
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  5. Pasin Manurangsi
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  6. Mikhail Rudoy
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  7. Anak Yodpinyanee
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Corresponding author

Correspondence to Jeffrey Bosboom .

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Editors and Affiliations

  1. Tokyo University of Science , Tokyo, Japan

    Jin Akiyama

  2. The University of Electro-Communications , Tokyo, Japan

    Hiro Ito

  3. Tokai University , Tokyo, Japan

    Toshinori Sakai

  4. Osaka Prefecture University , Sakai, Japan

    Yushi Uno

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Bosboom, J. et al. (2016). Dissection with the Fewest Pieces is Hard, Even to Approximate. In: Akiyama, J., Ito, H., Sakai, T., Uno, Y. (eds) Discrete and Computational Geometry and Graphs. JCDCGG 2015. Lecture Notes in Computer Science(), vol 9943. Springer, Cham. https://doi.org/10.1007/978-3-319-48532-4_4

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  • DOI: https://doi.org/10.1007/978-3-319-48532-4_4

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