On Polyatomic Tomography over Abelian Groups: Some Remarks on Consistency, Tree Packings and Complexity

Abstract

The paper deals with an inverse problem of reconstructing matrices from their marginal sums. More precisely, we are interested in the existence of 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. Keywords Discrete inverse problem • Discrete tomography • Polyatomic tomography • Polynomial-time algorithm • NP-completeness Mathematics Subject Classification Primary: 05 • 68R • 68Q17 • 82D • 90C • 49N45 Dedicated to the memory of Branko Grünbaum.