Dissection with the Fewest Pieces is Hard, Even to Approximate

  • Jeffrey Bosboom
  • Erik D. Demaine
  • Martin L. Demaine
  • Jayson Lynch
  • Pasin Manurangsi
  • Mikhail Rudoy
  • Anak Yodpinyanee
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, 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 \(1+1/1080-\varepsilon \).

References

  1. 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 CrossRefGoogle Scholar
  2. 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. 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. 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. 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 2007MathSciNetCrossRefMATHGoogle Scholar
  6. 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 4Google Scholar
  7. 7.
    Frederickson, G.N.: Dissections: Plane and Fancy. Cambridge University Press, Cambridge (1997)CrossRefMATHGoogle Scholar
  8. 8.
    Frederickson, G.N.: Hinged Dissections: Swinging & Twisting. Cambridge University Press, Cambridge (2002)MATHGoogle Scholar
  9. 9.
    Garey, M.R., Johnson, D.S.: Complexity results for multiprocessor scheduling under resource constraints. SIAM J. Comput. 4(4), 397–411 (1975)MathSciNetCrossRefMATHGoogle Scholar
  10. 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)MATHGoogle Scholar
  11. 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. IIIMathSciNetCrossRefGoogle Scholar
  12. 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. 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. 14.
    Wallace, W. (ed.): Elements of Geometry, 8th edn. Bell & Bradfute, Edinburgh (1831)Google Scholar

Copyright information

© Springer International Publishing AG 2016

Authors and Affiliations

  • Jeffrey Bosboom
    • 1
  • Erik D. Demaine
    • 1
  • Martin L. Demaine
    • 1
  • Jayson Lynch
    • 1
  • Pasin Manurangsi
    • 2
  • Mikhail Rudoy
    • 1
  • Anak Yodpinyanee
    • 1
  1. 1.Computer Science and AI LaboratoryMassachusetts Institute of TechnologyCambridgeUSA
  2. 2.University of CaliforniaBerkeleyUSA

Personalised recommendations