The paper deals with an inverse problem of reconstructing matrices from their marginal sums. More precisely, we are interested in the existence of $$r\times s$$ r × s matrices for which only the following information is available: The entries belong to known subsets of c distinguishable abelian groups, and the row and column sums of all entries from each group are given. This generalizes Ryser’s classical problem of characterizing the set of all 0–1-matrices with given row and column sums and is a basic problem in (polyatomic) discrete tomography. We show that the problem is closely related to packings of trees in bipartite graphs, prove consistency results, give algorithms and determine its complexity. In particular, we find a somewhat unusual complexity behavior: the problem is hard for “small” but easy for “large” matrices.

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关于阿贝尔群上的多原子层析成像：关于一致性、树填充和复杂性的一些评论

该论文涉及从矩阵的边际和重建矩阵的逆问题。更准确地说，我们对 $$r\times s$$ r × s 矩阵的存在感兴趣，其中只有以下信息可用：条目属于 c 个可区分阿贝尔群的已知子集，以及给出了每个组的所有条目。这概括了 Ryser 用给定的行和列总和来表征所有 0-1 矩阵的集合的经典问题，并且是（多原子）离散断层扫描中的一个基本问题。我们证明该问题与二部图中树的打包密切相关，证明了一致性结果，给出了算法并确定了其复杂性。特别是，我们发现了一个有点不寻常的复杂性行为：这个问题对“小”矩阵来说很难，但对“大”矩阵来说很容易。