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Title: Even 1xn Edge-Matching and Jigsaw Puzzles are Really Hard
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.  more » « less
Award ID(s):
1650733
NSF-PAR ID:
10078584
Author(s) / Creator(s):
; ; ; ; ;
Date Published:
Journal Name:
Journal of information processing
Volume:
25
ISSN:
1882-6652
Page Range / eLocation ID:
682–694
Format(s):
Medium: X
Sponsoring Org:
National Science Foundation
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