1 x 1 Rush Hour with Fixed Blocks is PSPACE-complete
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.
- Publication:
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arXiv e-prints
- Pub Date:
- March 2020
- DOI:
- 10.48550/arXiv.2003.09914
- arXiv:
- arXiv:2003.09914
- Bibcode:
- 2020arXiv200309914B
- Keywords:
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- Computer Science - Computational Complexity;
- Computer Science - Computational Geometry
- E-Print:
- 15 pages, 11 figures. Improved figures and writing. To appear at FUN 2020