1 x 1 Rush Hour with Fixed Blocks is PSPACE-complete

Josh Brunner; Lily Chung; Erik D. Demaine; Dylan Hendrickson; Adam Hesterberg; Adam Suhl; Avi Zeff

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arXiv · 2003.09914

1 x 1 Rush Hour with Fixed Blocks is PSPACE-complete

Josh Brunner·Lily Chung·Erik D. Demaine·Dylan Hendrickson·Adam Hesterberg·Adam Suhl·Avi Zeff

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Abstract

Consider $n^2-1$ unit-square blocks in an $n \times n$ square board, where each block is labeled as movable horizontally (only), movable vertically (only), or immovable -- a variation of Rush Hour with only $1 \times 1$ cars and fixed blocks. We prove that it is PSPACE-complete to decide whether a given block can reach the left edge of the board, by reduction from Nondeterministic Constraint Logic via 2-color oriented Subway Shuffle. By contrast, polynomial-time algorithms are known for deciding whether a given block can be moved by one space, or when each block either is immovable or can move both horizontally and vertically. Our result answers a 15-year-old open problem by Tromp and Cilibrasi, and strengthens previous PSPACE-completeness results for Rush Hour with vertical $1 \times 2$ and horizontal $2 \times 1$ movable blocks and 4-color Subway Shuffle.

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Josh Brunner, Lily Chung, Erik D. Demaine, Dylan Hendrickson, Adam Hesterberg, Adam Suhl, Avi Zeff. 2020-03-22. 1 x 1 Rush Hour with Fixed Blocks is PSPACE-complete. https://arxiv.org/abs/2003.09914

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