Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: 1 x 1 Rush Hour with Fixed Blocks is PSPACE-complete   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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The categories of this paper are: math
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The categories of this paper are: physics
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The categories of this paper are: alg-geom
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The categories of this paper are: plasm-ph
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The categories of this paper are: funct-an
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The categories of this paper are: dg-ga
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The categories of this paper are: patt-sol
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The categories of this paper are: supr-con
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The categories of this paper are: quant-ph
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The categories of this paper are: math-ph
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The categories of this paper are: econ
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The categories of this paper are: hep-th
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The categories of this paper are: hep-ph
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The categories of this paper are: solv-int
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The categories of this paper are: astro-ph
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The categories of this paper are: hep-lat