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

03/22/2020

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by   Josh Brunner, et al.

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Consider n^2-1 unit-square blocks in an n × 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 × 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 × 2 and horizontal 2 × 1 movable blocks and 4-color Subway Shuffle.