We analyze the computational complexity of Tetris clearing (determining whether the player can clear an initial board using a given sequence of pieces) and survival (determining whether the player can avoid losing before placing all the given pieces in an initial board) when restricted to a single polyomino piece type. We prove, for any tetromino piece type $P$ except for O, the NP-hardness of Tetris clearing and survival under the standard Super Rotation System (SRS), even when the input sequence consists of only a specified number of $P$ pieces. These surprising results disprove a 23-year-old conjecture on the computational complexity of Tetris with only I pieces (although our result is only for a specific rotation system). As a corollary, we prove the NP-hardness of Tetris clearing when the sequence of pieces has to be able to be generated from a $7k$-bag randomizer for any positive integer $k\geq 1$. On the positive side, we give polynomial-time algorithms for Tetris clearing and survival when the input sequence consists of only dominoes, assuming a particular rotation model, solving a version of a 9-year-old open problem. Along the way, we give polynomial-time algorithms for Tetris clearing and survival with $1\times k$ pieces (for any fixed $k$), provided the top $k-1$ rows are initially empty, showing that our I NP-hardness result needs to have filled cells in the top three rows.

翻译：我们分析了在仅限单一多联方块类型时，俄罗斯方块的消除游戏（判断玩家能否利用给定序列的方块清除初始棋盘）和生存游戏（判断玩家能否在放置完初始棋盘上所有给定方块前避免失败）的计算复杂性。我们证明，对于除O之外的任意四格方块类型$P$，在标准超级旋转系统（SRS）下，即使输入序列仅包含指定数量的$P$方块，俄罗斯方块的消除和生存问题均为NP难问题。这一惊人结果推翻了一个存在23年之久的猜想（该猜想关于仅用I型方块时俄罗斯方块的计算复杂性，尽管我们的结论仅针对特定旋转系统）。作为推论，我们证明了当方块序列必须由$7k$-bag随机生成器生成时（对任意正整数$k\geq 1$），俄罗斯方块的消除问题也是NP难的。另一方面，在特定旋转模型下，我们给出了当输入序列仅包含多米诺骨牌时，俄罗斯方块消除和生存问题的多项式时间算法，解决了存在9年的一个开放问题的部分版本。此外，我们针对$1\times k$型方块（对任意固定$k$）给出了俄罗斯方块消除和生存问题的多项式时间算法（前提是初始棋盘顶部$k-1$行为空），表明我们的I型方块NP难结论需要顶部三行存在填充单元格。