Reconstructing Polyatomic Structures from Discrete X-Rays: NP-Completeness Proof for Three Atoms
نویسندگان
چکیده
We address a discrete tomography problem that arises in the study of the atomic structure of crystal lattices. A polyatomic structure T can be defined as an integer lattice in dimension D ≥ 2, whose points may be occupied by c distinct types of atoms. To “analyze” T , we conduct l measurements that we call discrete X-rays. A discrete X-ray in direction ξ determines the number of atoms of each type on each line parallel to ξ. Given l such non-parallel X-rays, we wish to reconstruct T . The complexity of the problem for c = 1 (one atom type) has been completely determined by Gardner, Gritzmann and Prangenberg [5], who proved that the problem is NP-complete for any dimension D ≥ 2 and l ≥ 3 non-parallel X-rays, and that it can be solved in polynomial time otherwise [9]. The NP-completeness result above clearly extends to any c ≥ 2, and therefore when studying the polyatomic case we can assume that l = 2. As shown in another article by the same authors, [4], this problem is also NP-complete for c ≥ 6 atoms, even for dimension D = 2 and axis-parallel X-rays. The authors of [4] conjecture that the problem remains NP-complete for c = 3, 4, 5, although, as they point out, the proof idea in [4] does not seem to extend to c ≤ 5. We resolve the conjecture from [4] by proving that the problem is indeed NP-complete for c ≥ 3 in 2D, even for axis-parallel X-rays. Our construction relies heavily on some structure results for the realizations of 0-1 matrices with given row and column sums.
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