Tetris is Hard with Just One Piece Type

TL;DR

This paper proves the NP-hardness of Tetris clearing and survival with most single polyomino pieces under standard rules, while providing polynomial algorithms for certain simpler cases, thus resolving longstanding open problems.

Contribution

It establishes NP-hardness for Tetris with all but one piece type under standard rules and offers polynomial algorithms for specific restricted cases.

Findings

NP-hardness of Tetris with P (except O) under SRS

Polynomial algorithms for dominoes with certain assumptions

NP-hardness of Tetris with sequences from a 7k-bag randomizer

Abstract

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.