Even $1 \times n$ 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-hard even to approximately maximize the number of placed tiles (allowing blanks), while satisfying the compatibility constraint between nonblank tiles, within a factor of 0.9999999851. (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 NPhardness 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.