  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-epsilon), when given a sequence of p
pieces, and the inapproximability of the third objective to within a factor of
(2 - epsilon), for any epsilon>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.
