Tetris is Hard, Even to Approximate [ps | ps.gz | pdf ]

Ron Breukelaar, Erik D. Demaine, Susan Hohenberger, Hendrik Jan Hoogeboom, Walter A. Kosters, and David Liben-Nowell

In International Journal of Computational Geometry and Applications, 14(1-2):41-68, April 2004.
A previous version of parts of this paper appears in Proceedings of COCOON 2003 by Erik D. Demaine, Susan Hohenberger, and David Liben-Nowell.
A two-page version (ps, pdf) appears in Proceedings of Fall Workshop on Computational Geometry '02 by the same authors.

The most recent update to this document was on 04 August 2005.

In the popular computer game of Tetris, the player is given a sequence of tetromino pieces and must pack them into a rectangular gameboard initially occupied by a given configuration of filled squares; any completely filled row of the gameboard is cleared and all pieces above it drop by one row. We prove that in the offline version of Tetris, it is NP-complete to maximize the number of cleared rows, maximize the number of tetrises (quadruples of rows simultaneously filled and cleared), minimize the maximum height of an occupied square, or maximize the number of pieces placed before the game ends. We furthermore show the extreme inapproximability of the first and last of these objectives to within a factor of p^(1-ε), when given a sequence of p pieces, and the inapproximability of the third objective to within a factor of (2 - ε), for any ε>0. Our results hold under several variations on the rules of Tetris, including different models of rotation, limitations on player agility, and restricted piece sets.

  author = 	 {Ron Breukelaar 
                  and Erik D.~Demaine
                  and Susan Hohenberger
                  and Hendrik Jan Hoogeboom
                  and Walter A.~Kosters
                  and David Liben-Nowell},
  title = 	 {Tetris is Hard, Even to Approximate},
  journal = 	 {International Journal of Computational Geometry
                  and Applications},
  year = 	 2004,
  volume =	 14,
  number =	 {1--2},
  pages =	 {41--68},
  month =	 apr,

David Liben-Nowell

04 August 2005