Even 1 × <i>n</i> Edge-Matching and Jigsaw Puzzles are Really Hard

Abstract

Abstract: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.

Edge Matching with Inequalities, Triangles, Unknown Shape, and Two Players

“…The conclusion of Ref. [7] claimed that the paper's results extended to equilateral-triangle edge matching, but the proposed simulation of squares by triangles is incorrect because it constrains the orientation of the simulated squares. In Section 4.1, we extend our 1 × n parsimonious proof to obtain NP/#P/ASPcompleteness for signed and unsigned edge matching with equilateral triangles, with or without reflection.…”

“…In 2007, Demaine and Demaine [10] proved that signed and unsigned edge-matching square-tile puzzles are NP-complete and equivalent to both jigsaw puzzles and polyomino packing puzzles. In 2016, Bosboom et al [7] proved that signed and unsigned edge-matching square-tile puzzles are NP-complete even when the target shape is a 1 × n rectangle, and furthermore hard to approximate within some constant factor. Our work on 1 × n triangle edge-matching puzzles is inspired by an open problem proposed in the latter paper.…”

“…In this section, we adapt the work of Ref. [7] to show that 1 × n edge-matching puzzles are ASP-and #P-complete. Like [7], we reduce from Hamiltonian path in planar 3-regular directed graphs, which we newly prove ASP-and #P-complete.…”

“…The reduction in Ref. [7] that establishes NP-hardness of 1 × n edge-matching puzzles is not parsimonious because of garbage collection. Namely, the tiles corresponding to edges which are not part of the Hamiltonian path are placed at the end of the row of tiles in an arbitrary order.…”

A Linear Threshold for Uniqueness of Solutions to Random Jigsaw Puzzles

“…Concerning the problem of how to recover the planted assembly efficiently, Bordenave et al describe an algorithm that recovers it with high probability when q ≥ n 1+ε with time complexity n O(1/ε) . As a comparison, the general problems of finding one solution to a given n × n jigsaw puzzle or edgematching puzzle are known to be NP-complete [4], see also [3]. The problem also seems to be hard in practice.…”