Even 1 × n 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-har...

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Main Authors: Bosboom, Jeffrey William, Demaine, Erik D, Demaine, Martin L, Hesterberg, Adam Classen, Manurangsi, Pasin, Yodpinyanee, Anak
Other Authors: Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science
Format: Article
Language:English
Published: Information Processing Society of Japan (Jōhō Shori Gakkai) 2019
Online Access:https://hdl.handle.net/1721.1/122826
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author Bosboom, Jeffrey William
Demaine, Erik D
Demaine, Martin L
Hesterberg, Adam Classen
Manurangsi, Pasin
Yodpinyanee, Anak
author2 Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science
author_facet Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science
Bosboom, Jeffrey William
Demaine, Erik D
Demaine, Martin L
Hesterberg, Adam Classen
Manurangsi, Pasin
Yodpinyanee, Anak
author_sort Bosboom, Jeffrey William
collection MIT
description 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
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spelling mit-1721.1/1228262022-09-28T00:57:00Z Even 1 × n Edge-Matching and Jigsaw Puzzles are Really Hard Bosboom, Jeffrey William Demaine, Erik D Demaine, Martin L Hesterberg, Adam Classen Manurangsi, Pasin Yodpinyanee, Anak 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 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 2019-11-12T01:19:08Z 2019-11-12T01:19:08Z 2017-08 2019-06-17T21:03:03Z Article http://purl.org/eprint/type/JournalArticle 1882-6652 https://hdl.handle.net/1721.1/122826 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 en https://doi.org/10.2197/ipsjjip.25.682 Journal of Information Processing Creative Commons Attribution-Noncommercial-Share Alike http://creativecommons.org/licenses/by-nc-sa/4.0/ application/pdf Information Processing Society of Japan (Jōhō Shori Gakkai) arXiv
spellingShingle Bosboom, Jeffrey William
Demaine, Erik D
Demaine, Martin L
Hesterberg, Adam Classen
Manurangsi, Pasin
Yodpinyanee, Anak
Even 1 × n Edge-Matching and Jigsaw Puzzles are Really Hard
title Even 1 × n Edge-Matching and Jigsaw Puzzles are Really Hard
title_full Even 1 × n Edge-Matching and Jigsaw Puzzles are Really Hard
title_fullStr Even 1 × n Edge-Matching and Jigsaw Puzzles are Really Hard
title_full_unstemmed Even 1 × n Edge-Matching and Jigsaw Puzzles are Really Hard
title_short Even 1 × n Edge-Matching and Jigsaw Puzzles are Really Hard
title_sort even 1 n edge matching and jigsaw puzzles are really hard
url https://hdl.handle.net/1721.1/122826
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