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Register a new userTetris is NP-hard even with $O(1)$ rows or columns
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Added by
Erik Demaine
Publication date
2020
fields
Informatics Engineering
and research's language is
English
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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.
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We prove that deciding whether the Runner can win this turn (mate-in-1) in the Netrunner card game generalized to allow decks to contain an arbitrary number of copies of a card is weakly NP-hard. We also prove that deciding whether the Corp can win within two turns (mate-in-2) in this generalized Netrunner is weakly NP-hard.
The computational complexity of a problem arising in the context of sparse optimization is considered, namely, the projection onto the set of $k$-cosparse vectors w.r.t. some given matrix $Omeg$. It is shown that this projection problem is (strongly) NP-hard, even in the special cases in which the matrix $Omeg$ contains only ternary or bipolar coefficients. Interestingly, this is in contrast to the projection onto the set of $k$-sparse vectors, which is trivially solved by keeping only the $k$ largest coefficients.
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We prove PSPACE-completeness of two classic types of Chess problems when generalized to n-by-n boards. A retrograde problem asks whether it is possible for a position to be reached from a natural starting position, i.e., whether the position is valid or legal or reachable. Most real-world retrograde Chess problems ask for the last few moves of such a sequence; we analyze the decision question which gets at the existence of an exponentially long move sequence. A helpmate problem asks whether it is possible for a player to become checkmated by any sequence of moves from a given position. A helpmate problem is essentially a cooperative form of Chess, where both players work together to cause a particular player to win; it also arises in regular Chess games, where a player who runs out of time (flags) loses only if they could ever possibly be checkmated from the current position (i.e., the helpmate problem has a solution). Our PSPACE-hardness reductions are from a variant of a puzzle game called Subway Shuffle.
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