Even 1 × n Edge-Matching and Jigsaw Puzzles are Really Hard
Name
1701.00146.pdf
Description
Submitted version
Size
328.21 KB
Format
Adobe PDF
Checksum (MD5)
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Author(s)
Bosboom, Jeffrey William
Demaine, Erik D
Demaine, Martin L
Hesterberg, Adam Classen
Manurangsi, Pasin
Yodpinyanee, Anak
Date Issued
August 2017
Journal
Journal of Information Processing
Publisher
Information Processing Society of Japan (Jōhō Shori Gakkai)
Citation
Jeffrey Bosboom, Erik D. Demaine, Martin L. Demaine, Adam Hesterberg, Pasin Manurangsi, Anak Yodpinyanee. "Even 1 × n Edge-Matching and Jigsaw Puzzles are Really Hard." Journal of Information Processing, 25 (August 2017): 682-694 © 2017 Information Processing Society of Japan
Version
Original manuscript
Abstract
We prove the computational intractability of rotating and placing n square tiles into a 1 × n array such that adjacent tiles are compatible-either equal edge colors, as in edge-matching puzzles, or matching tab/pocket shapes, as in jigsaw puzzles. Beyond basic NP-hardness, we prove that it is NP-hard even to approximately maximize the number of placed tiles (allowing blanks), while satisfying the compatibility constraint between nonblank tiles, within a factor of 0.9999999702 (On the other hand, there is an easy 1/2 -approximation). This is the first (correct) proof of inapproximability for edge-matching and jigsaw puzzles. Along the way, we prove NP-hardness of distinguishing, for a directed graph on n nodes, between having a Hamiltonian path (length n - 1) and having at most 0.999999284(n - 1) edges that form a vertex-disjoint union of paths. We use this gap hardness and gap-preserving reductions to establish similar gap hardness for 1 × n jigsaw and edge-matching puzzles. Keywords: edge-matching puzzles; jigsaw puzzles; computational complexity; hardness of approximation
MIT Department
Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science
Massachusetts Institute of Technology. Department of Materials Science and Engineering
Massachusetts Institute of Technology. Computer Science and Artificial Intelligence Laboratory
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Creative Commons Attribution-Noncommercial-Share Alike
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DOI of Published Version
https://doi.org/10.2197/ipsjjip.25.682
