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...
Main Authors: | , , , , , |
---|---|
Other Authors: | |
Format: | Article |
Language: | English |
Published: |
Information Processing Society of Japan (Jōhō Shori Gakkai)
2019
|
Online Access: | https://hdl.handle.net/1721.1/122826 |
_version_ | 1811083015439253504 |
---|---|
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 |
first_indexed | 2024-09-23T12:18:00Z |
format | Article |
id | mit-1721.1/122826 |
institution | Massachusetts Institute of Technology |
language | English |
last_indexed | 2024-09-23T12:18:00Z |
publishDate | 2019 |
publisher | Information Processing Society of Japan (Jōhō Shori Gakkai) |
record_format | dspace |
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 |
work_keys_str_mv | AT bosboomjeffreywilliam even1nedgematchingandjigsawpuzzlesarereallyhard AT demaineerikd even1nedgematchingandjigsawpuzzlesarereallyhard AT demainemartinl even1nedgematchingandjigsawpuzzlesarereallyhard AT hesterbergadamclassen even1nedgematchingandjigsawpuzzlesarereallyhard AT manurangsipasin even1nedgematchingandjigsawpuzzlesarereallyhard AT yodpinyaneeanak even1nedgematchingandjigsawpuzzlesarereallyhard |