2016/12/31 by Bosboom, Jeffrey, Demaine, Erik D., Demaine, Martin L. +3
#Computational Complexity (cs.CC) #Computational Geometry (cs.CG) #FOS: Computer and information sciences
paper · doi:10.48550/arxiv.1701.00146
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 \over 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.