Reduction from three-dimensional discrete tomography to multicommodity flow problem

The reduction from two-dimensional-discrete tomography to max-flow problem is well-known [Gale, A theorem on flows in networks, Pacific J. Math. 7 (1957) 1073-1082]. This approach is based on the natural correspondence between two-dimensional lattices and bipartite graphs. We extend this result in dimension 3 by reducing three-dimensional discrete tomography to multicommodity flow problems. Two reductions are presented, one considering discrete tomography with multisets while the other one works with sets.