Paper by Martin L. Demaine
- Jeffrey Bosboom, Erik D. Demaine, Martin L. Demaine, Adam Hesterberg, Pasin Manurangsi, and Anak Yodpinyanee, “Even 1 × n Edge Matching and Jigsaw Puzzles are Really Hard”, Journal of Information Processing, volume 25, 2017, pages 682–694. Special issue of papers from the 19th Japan Conference on Discrete and Computational Geometry, Graphs, and Games
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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- The paper is 11 pages.
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- Related papers:
- Jigsaw1xn_JCDCGGG2016 (Even 1 × n Edge Matching and Jigsaw Puzzles are Really Hard)
See also other papers by Martin Demaine.
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