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arXiv · 2009.14336

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

Sualeh Asif·Michael Coulombe·Erik D. Demaine·Martin L. Demaine·Adam Hesterberg·Jayson Lynch·Mihir Singhal
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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.

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Sualeh Asif, Michael Coulombe, Erik D. Demaine, Martin L. Demaine, Adam Hesterberg, Jayson Lynch, Mihir Singhal. 2020-09-29. Tetris is NP-hard even with $O(1)$ rows or columns. https://arxiv.org/abs/2009.14336

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