Combinatorial optimization: algorithms and complexity
Combinatorial optimization: algorithms and complexity
Sets uniquely determined by projections on axes II Discrete case
Discrete Mathematics
Reductions of additive sets, sets of uniqueness and pyramids
Discrete Mathematics
Cancellation conditions for finite two-dimensional additive measurement
Journal of Mathematical Psychology
On the algorithmic inversion of the discrete Radon transform
Theoretical Computer Science
A characterization of additive sets
Discrete Mathematics
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In Discrete Tomography there are two related notions of interest: H-uniqueness and H-additivity of finite subsets of N^m, which are defined for certain finite sets H of linear subspaces of R^m. One knows complete sets of obstructions for H-uniqueness (bad H-configurations) and for H-additivity (weakly bad H-configurations). The classical case, when H is the set of coordinate axes in R^2, is well known. Let H"m denote the set of the m coordinate hyperplanes of R^m. The following question was raised in [P.C. Fishburn, J.C. Lagarias, J.A. Reeds, L.A. Shepp, Sets uniquely determined by projections on axes II. Discrete case, Discrete Math. 91 (1991) 149-159]. Is there an upper bound on the weights of the bad H"m-configurations one needs to consider to determine H"m-uniqueness (m=3) of an arbitrary set in N^m? This question can be asked for other sets H of linear subspaces and also for H-additivity. The answer to this question, in the case of uniqueness, is known when H is a set of lines. In this paper we answer this question for uniqueness and additivity in the case of H"3. We show that there is no upper bound on the weights of the bad configurations (resp. weakly bad configurations) one needs to consider to determine H"3-uniqueness (resp. H"3-additivity).