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NP-hardness under the 7-bag randomizer

Show NP-hardness (or otherwise characterize the computational complexity) of Tetris clearing under the Super Rotation System (SRS) when the input sequence must be generated by a 7-bag randomizer, i.e., partitioned into consecutive bags each containing exactly one of each tetromino in some permutation.

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Background

All reductions in the paper assume an arbitrary fixed sequence of pieces. Many modern Tetris implementations restrict randomness using a 7-bag generator, which produces a sequence as a concatenation of permutations of all seven tetrominoes.

Proving NP-hardness (or otherwise classifying complexity) under this additional restriction would align theoretical results with common modern gameplay constraints and requires reductions robust to bag constraints.

References

Some open questions arise regarding whether our results can be extended if we consider different objectives or add additional features. Modern variants of Tetris also use different random generators to ensure that the player does not receive the same piece arbitrarily many times in a row. One of the simplest random generators is called a \defn{7-bag randomizer}. For this randomizer, the sequence of pieces is divided into groups (or "bags") of 7, each group containing one of each tetromino in one of $7! = 5{,}040$ possible orderings. Can we show NP-hardness even if the sequence of pieces has to be able to be generated from this randomizer?

— Tetris with Few Piece Types (2404.10712 - Group et al., 16 Apr 2024) in Section 7 (Open Problems)