Document Open Access Logo

Tetris with Few Piece Types

Authors MIT Hardness Group, Erik D. Demaine , Holden Hall, Jeffery Li

PDF
Thumbnail PDF

File

LIPIcs.FUN.2024.24.pdf
  • Filesize: 1.06 MB
  • 18 pages

Document Identifiers

Author Details

MIT Hardness Group
  • CSAIL, Massachusetts Institute of Technology, Cambridge, MA, USA
Erik D. Demaine
  • CSAIL, Massachusetts Institute of Technology, Cambridge, MA, USA
Holden Hall
  • CSAIL, Massachusetts Institute of Technology, Cambridge, MA, USA
Jeffery Li
  • CSAIL, Massachusetts Institute of Technology, Cambridge, MA, USA

Acknowledgements

This paper was initiated during open problem solving in the MIT class on Algorithmic Lower Bounds: Fun with Hardness Proofs (6.5440) taught by Erik Demaine in Fall 2023. We thank the other participants of that class for helpful discussions and providing an inspiring atmosphere. Figures drawn with SVG Tiler (https://github.com/edemaine/svgtiler).

Cite AsGet BibTex

MIT Hardness Group, Erik D. Demaine, Holden Hall, and Jeffery Li. Tetris with Few Piece Types. In 12th International Conference on Fun with Algorithms (FUN 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 291, pp. 24:1-24:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.FUN.2024.24

Abstract

We prove NP-hardness and #P-hardness of Tetris clearing (clearing an initial board using a given sequence of pieces) with the Super Rotation System (SRS), even when the pieces are limited to any two of the seven Tetris piece types. This result is the first advance on a question posed twenty years ago: which piece sets are easy vs. hard? All previous Tetris NP-hardness proofs used five of the seven piece types. We also prove ASP-completeness of Tetris clearing, using three piece types, as well as versions of 3-Partition and Numerical 3-Dimensional Matching where all input integers are distinct. Finally, we prove NP-hardness of Tetris survival and clearing under the "hard drops only" and "20G" modes, using two piece types, improving on a previous "hard drops only" result that used five piece types.

Subject Classification

ACM Subject Classification
  • Theory of computation → Problems, reductions and completeness
Keywords
  • complexity
  • hardness
  • video games
  • counting

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads
    0
    Metadata Views

Related Versions

References

  1. Tetris, 2023. Apple TV+ movie directed by Jon S. Baird and written by Noah Pink. Google Scholar
  2. Sualeh Asif, Michael Coulombe, Erik D. Demaine, Martin L. Demaine, Adam Hesterberg, Jayson Lynch, and Mihir Singhal. Tetris is NP-hard even with O(1) rows or columns. Journal of Information Processing, 28:942-958, 2020. Google Scholar
  3. Jeffrey Bosboom, Erik D. Demaine, Martin L. Demaine, Adam Hesterberg, Roderick Kimball, and Justin Kopinsky. Path puzzles: Discrete tomography with a path constraint is hard. Graphs and Combinatorics, 36:251-267, 2020. Google Scholar
  4. Ron Breukelaar, Erik D. Demaine, Susan Hohenberger, Hendrik Jan Hoogeboom, Walter A. Kosters, and David Liben-Nowell. Tetris is hard, even to approximate. International Journal of Computational Geometry and Applications, 14(1-2):41-68, 2004. URL: https://doi.org/10.1142/S0218195904001354.
  5. Charles J. Colbourn. The complexity of completing partial Latin squares. Discrete Applied Mathematics, 8(1):25-30, 1984. URL: https://doi.org/10.1016/0166-218X(84)90075-1.
  6. Sopan Deb. Boy, 13, is believed to be the first to ‘beat’ tetris. The New York Times, 3 january 2024. URL: https://www.nytimes.com/2024/01/03/arts/tetris-beat-blue-scuti.html.
  7. Erik D. Demaine, Martin L. Demaine, Sarah Eisenstat, Adam Hesterberg, Andrea Lincoln, Jayson Lynch, and Y. William Yu. Total Tetris: Tetris with monominoes, dominoes, trominoes, pentominoes, .... Journal of Information Processing, 25:515-527, 2017. URL: https://doi.org/10.2197/ipsjjip.25.515.
  8. Erik D. Demaine, Fermi Ma, Erik Waingarten, Ariel Schvartzman, and Scott Aaronson. The fewest clues problem. Theoretical Computer Science, 748:28-39, November 2018. URL: https://doi.org/10.1016/j.tcs.2018.01.020.
  9. Ian Holyer. The NP-completeness of some edge-partition problems. SIAM Journal on Computing, 10(4):713-717, 1981. URL: https://doi.org/10.1137/0210054.
  10. Heather Hulett, Todd G. Will, and Gerhard J. Woeginger. Multigraph realizations of degree sequences: Maximization is easy, minimization is hard. Operations Research Letters, 36(5):594-596, 2008. URL: https://doi.org/10.1016/j.orl.2008.05.004.
  11. Harry B. Hunt III, Madhav V. Marathe, Venkatesh Radhakrishnan, and Richard E. Stearns. The complexity of planar counting problems. SIAM Journal on Computing, 27(4):1142-1167, 1998. URL: https://doi.org/10.1137/S0097539793304601.
  12. Oscar Temprano. Complexity of a Tetris variant. https://arxiv.org/abs/1506.07204, 2015. URL: https://arxiv.org/abs/1506.07204,
  13. TetrisWiki. 20G. https://tetris.wiki/20G. Accessed: 2023-09-23.
  14. TetrisWiki. Super Rotation System. https://tetris.wiki/Super_Rotation_System. Accessed: 2023-09-23.
  15. TetrisWiki. Tetris guideline. https://tetris.wiki/Tetris_Guideline. Accessed: 2023-09-23.
  16. Wikipedia. Tetris. https://en.wikipedia.org/wiki/Tetris. Accessed: 2023-09-23.
  17. Takayuki Yato and Takahiro Seta. Complexity and completeness of finding another solution and its application to puzzles. IEICE Transactions on Fundamentals of Electronics, Communications, and Computer Sciences, E86-A(5):1052-1060, 2003. Also IPSJ SIG Notes 2002-AL-87-2, 2002. URL: http://ci.nii.ac.jp/naid/110003221178/en/.
© 2023-2024 Schloss Dagstuhl – LZI GmbH Imprint Privacy Contact