Complexity of tile rotation problems
Theoretical Computer Science
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The complexity of satisfiability problems
STOC '78 Proceedings of the tenth annual ACM symposium on Theory of computing
Jigsaw Puzzles, Edge Matching, and Polyomino Packing: Connections and Complexity
Graphs and Combinatorics
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IJCAI'91 Proceedings of the 12th international joint conference on Artificial intelligence - Volume 1
Approximability of Edge Matching Puzzles
SOFSEM '10 Proceedings of the 36th Conference on Current Trends in Theory and Practice of Computer Science
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Edge-matching problems, also called puzzles, are abstractions of placement problems with neighborhood conditions. Pieces with colored edges have to be placed on a board such that adjacent edges have the same color. The problem has gained interest recently with the (now terminated) Eternity II puzzle, and new complexity results. In this paper we consider a number of settings which differ in size of the puzzles and the manipulations allowed on the pieces. We investigate the effect of allowing rotations of the pieces on the complexity of the problem, an aspect that is only marginally treated so far. We show that some problems have polynomial time algorithms while others are NP-complete. Especially we show that allowing rotations in one-row puzzles makes the problem NP-hard. We moreover show that many commonly considered puzzles can be emulated by simple puzzles with quadratic pieces, so that one can restrict oneself to investigating those.