Journal of Information Processing
Online ISSN : 1882-6652
Even 1 × n Edge-Matching and Jigsaw Puzzles are Really Hard
Jeffrey BosboomErik D. DemaineMartin L. DemaineAdam HesterbergPasin ManurangsiAnak Yodpinyanee
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Volume 25 (2017) Pages 682-694

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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.

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© 2017 by the Information Processing Society of Japan
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