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Japanese Conference on Discrete and Computational Geometry and Graphs

JCDCGG 2015: Discrete and Computational Geometry and Graphs pp 37–48Cite as

Dissection with the Fewest Pieces is Hard, Even to Approximate

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

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.

References

  1. Aloupis, G., Demaine, E.D., Demaine, M.L., Dujmović, V., Iacono, J.: Meshes preserving minimum feature size. In: Márquez, A., Ramos, P., Urrutia, J. (eds.) EGC 2011. LNCS, vol. 7579, pp. 258–273. Springer, Heidelberg (2012). doi:10.1007/978-3-642-34191-5_25

    Chapter  Google Scholar 

  2. Bolyai, F.: Tentamen juventutem studiosam in elementa matheseos purae, elementaris ac sublimioris, methodo intuitiva, evidentiaque huic propria, introducendi. Typis Collegii Refomatorum per Josephum et Simeonem Kali, Maros Vásárhely (1832–1833)

    Google Scholar 

  3. Bosboom, J., Demaine, E.D., Demaine, M.L., Lynch, J., Manurangsi, P., Rudoy, M., Yodpinyanee, A.: Dissection with the fewest pieces is hard, even to approximate. CoRR abs/1512.06706 (2015). http://arxiv.org/abs/1512.06706

  4. Canny, J.: Some algebraic and geometric computations in PSPACE. In: Proceedings of the Twentieth Annual ACM Symposium on Theory of Computing, STOC 1988, pp. 460–467. ACM, New York (1988). http://doi.acm.org/10.1145/62212.62257

  5. Demaine, E.D., Demaine, M.L.: Jigsaw puzzles, edge matching, and polyomino packing: connections and complexity. Graphs Comb. 23(Suppl.), 195–208 (2007). Special issue on Computational Geometry and Graph Theory: The Akiyama-Chvatal Festschrift. Preliminary version presented at KyotoCGGT 2007

    Article  MathSciNet  MATH  Google Scholar 

  6. Dudeney, H.E.: Puzzles and prizes. Weekly Dispatch (1902), the puzzle appeared in the April 6 issue of this column. A discussion followed on April 20, and the solution appeared on May 4

    Google Scholar 

  7. Frederickson, G.N.: Dissections: Plane and Fancy. Cambridge University Press, Cambridge (1997)

    Book  MATH  Google Scholar 

  8. Frederickson, G.N.: Hinged Dissections: Swinging & Twisting. Cambridge University Press, Cambridge (2002)

    MATH  Google Scholar 

  9. Garey, M.R., Johnson, D.S.: Complexity results for multiprocessor scheduling under resource constraints. SIAM J. Comput. 4(4), 397–411 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  10. Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman & Co., New York (1979)

    MATH  Google Scholar 

  11. Gerwien, P.: Zerschneidung jeder beliebigen Anzahl von gleichen geradlinigen Figuren in dieselben Stücke. Journal für die reine und angewandte Mathematik (Crelle’s Journal) 10, 228–234 (1833). Taf. III

    Article  MathSciNet  Google Scholar 

  12. Hazan, E., Safra, S., Schwartz, O.: On the hardness of approximating k-dimensional matching. Electronic Colloquium on Computational Complexity (ECCC) 10(020) (2003). http://eccc.hpi-web.de/eccc-reports/2003/TR03-020/index.html

  13. Lowry, M.: Solution to question 269, [proposed] by Mr. W. Wallace. In: Leybourn, T. (ed.) Mathematical Repository Part 1, pp. 44–46. W. Glendinning, London (1814)

    Google Scholar 

  14. Wallace, W. (ed.): Elements of Geometry, 8th edn. Bell & Bradfute, Edinburgh (1831)

    Google Scholar 

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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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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-48531-7

  • Online ISBN: 978-3-319-48532-4

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