License: Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.FUN.2021.7
URN: urn:nbn:de:0030-drops-127681
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Brunner, Josh ; Chung, Lily ; Demaine, Erik D. ; Hendrickson, Dylan ; Hesterberg, Adam ; Suhl, Adam ; Zeff, Avi

1 X 1 Rush Hour with Fixed Blocks Is PSPACE-Complete

LIPIcs-FUN-2021-7.pdf (0.7 MB)


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 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.

BibTeX - Entry

  author =	{Josh Brunner and Lily Chung and Erik D. Demaine and Dylan Hendrickson and Adam Hesterberg and Adam Suhl and Avi Zeff},
  title =	{{1 X 1 Rush Hour with Fixed Blocks Is PSPACE-Complete}},
  booktitle =	{10th International Conference on Fun with Algorithms (FUN 2021)},
  pages =	{7:1--7:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-145-0},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{157},
  editor =	{Martin Farach-Colton and Giuseppe Prencipe and Ryuhei Uehara},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{},
  URN =		{urn:nbn:de:0030-drops-127681},
  doi =		{10.4230/LIPIcs.FUN.2021.7},
  annote =	{Keywords: puzzles, sliding blocks, PSPACE-hardness}

Keywords: puzzles, sliding blocks, PSPACE-hardness
Collection: 10th International Conference on Fun with Algorithms (FUN 2021)
Issue Date: 2020
Date of publication: 16.09.2020

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