On multi-index assignment polytopes

We investigate an integer programming model for multi-dimensional assignment problems. This model enables us to establish the dimension for entire families of assignment polytopes, thus unifying and generalising previous results. In particular, we establish the dimension of the linear assignment polytope as well as that of every axial and planar assignment polytope. Further, for the axial polytopes, we identify a family of clique facets. We also give a necessary condition for the existence of a solution for assignment problems.