Tetris is NP-hard even with $O(1)$ rows or columns

Abstract

We prove that the classic falling-block video game Tetris (both survival and board clearing) remains NP-complete even when restricted to 8 columns, or to 4 rows, settling open problems posed over 15 years ago [BDH + 04]. Our reduction is from 3-Partition, similar to the previous reduction for unrestricted board sizes, but with a better packing of buckets. On the positive side, we prove that 2-column Tetris (and 1-row Tetris) is polynomial. We also prove that the generalization of Tetris to larger k-omino pieces is NP-complete even when the board starts empty, even when restricted to 3 columns or 2 rows or constant-size pieces. Finally, we present an animated Tetris font.

How fast can we play Tetris greedily with rectangular pieces?

“…This work was generalized to show NP-completeness for k-ominos with k ≥ 4, and surprisingly also show that it is NP-complete to clear the board even for k = 2 when rotations are not allowed [34]. These reductions were further tightened in [11,12] to when the board is small and in [89] for the variant where you can rotate and then move down, but when no rotations are allowed after the block has begun to fall. The undecidability of whether a set of sequences of Tetris moves described by a regular language contains a sequence that results in an empty board has been proven [46].…”

How Fast Can We Play Tetris Greedily With Rectangular Pieces?

“…This work was generalized to shown NP completes for k-ominos other than 4, and surprisingly showed that it is NP-complete to clear the board even for k = 2 when rotations not allowed [43]. These reductions were further tightened in [13,12] to when the board is small and in [141] for the variant where you can rotate and then move down, but when no rotations are allowed after the block has begun to fall. The indecidability of whether a set of sequences of Tetris moves described by a regular language contains a sequence that results in an empty board [64].…”