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Tetris is NP-hard even with $O(1)$ rows or columns
arXiv - CS - Computational Complexity Pub Date : 2020-09-29 , DOI: arxiv-2009.14336
Sualeh Asif, Michael Coulombe, Erik D. Demaine, Martin L. Demaine, Adam Hesterberg, Jayson Lynch, Mihir Singhal

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.

中文翻译:

即使有 $O(1)$ 行或列,俄罗斯方块也是 NP-hard

我们证明,即使限制为 8 列或 4 行,经典的落块视频游戏俄罗斯方块(生存和棋盘清理)仍然是 NP 完全的,解决了 15 年前提出的开放问题 [BDH+04]。我们的减少来自 3-Partition,类似于之前对不受限制的电路板尺寸的减少,但具有更好的桶包装。从积极的方面来说,我们证明了 2 列俄罗斯方块(和 1 行俄罗斯方块)是多项式。我们还证明,即使棋盘开始为空,即使限制为 3 列或 2 行或恒定大小的棋子,俄罗斯方块对更大的 $k$-omino 棋子的泛化也是 NP 完全的。最后,我们展示了一个动画俄罗斯方块字体。
更新日期:2020-10-01
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