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Discrete tomography deals with reconstructing finite spatial objects from their projections. The objects we study in this paper are called tilings or tile-packings, and they consist of a number of disjoint copies of a fixed tile, where a tile is defined as a connected set of grid
points. A row projection specifies how many grid points are covered by tiles in a given row; column projections are defined analogously. For a fixed tile, is it possible to reconstruct its tilings from their projections in polynomial time? It is known that the answer to this question is affirmative if the tile is a bar (its width or height is 1), while for some other types of tiles NP-hardness results have been shown in the literature. In this paper we present a complete solution to this question by showing that the problem remains NP-hard for all tiles other than bars.
CitationChrobak, M. [et al.]. Tile-packing tomography is NP-hard. A: International Computing and Combinatorics Conference. "Computing and Combinatorics: 16th Annual International Conference, COCOON 2010: Nha Trang, Vietnam, July 19-21, 2010: Proceedings". Nha Trang: 2010, p. 254-263.
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