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, Ryuhei Uehara, Martin Farach-Colton, Giuseppe Prencipe
2020 Fun with Algorithms  
Consider n²-1 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
more » ... lock 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.
doi:10.4230/lipics.fun.2021.7 dblp:conf/fun/BrunnerCDHHSZ21 fatcat:xanal3nqszhyleccjmaurumd6i