The paper proves that Tetris clearing is NP-hard and #P-hard even when the piece types are restricted to any size-2 subset of the seven Tetris piece types. The key insights are:
For every size-2 subset of piece types, the authors provide a reduction from 3-Partition with Distinct Integers to Tetris clearing. The reduction involves constructing "bottles" with a neck portion and a body portion, and using a sequence of pieces to properly block all but one bottle, fill the unblocked bottle, and then reset the state of the bottles.
The authors also prove that 3-Partition with Distinct Integers and Numerical 3-Dimensional Matching with Distinct Integers are strongly ASP-complete, which is a stronger result than the previous strong NP-completeness proofs.
For certain size-3 subsets of piece types, the authors further establish ASP-completeness of Tetris clearing, which implies NP-hardness of finding another solution given k solutions, for any k≥0, as well as #P-completeness.
The authors also study Tetris under the "hard drops only" and "20G" models, and establish NP-hardness of both Tetris survival and clearing for certain size-2 subsets of piece types under these models.
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arxiv.org
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