Even 1 × n Edge-Matching and Jigsaw Puzzles are Really Hard

• Main Authors: Bosboom, Jeffrey William (Author), Demaine, Erik D (Author), Demaine, Martin L (Author), Hesterberg, Adam Classen (Author), Manurangsi, Pasin (Author), Yodpinyanee, Anak (Author)
• Other Authors: Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science (Contributor), Massachusetts Institute of Technology. Department of Materials Science and Engineering (Contributor), Massachusetts Institute of Technology. Computer Science and Artificial Intelligence Laboratory (Contributor)
• Format: Article
• Language: English
• Published: Information Processing Society of Japan (Jōhō Shori Gakkai), 2019-11-12T01:19:08Z.
• Subjects: Article
• Online Access: Get fulltext

 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