Polyhedral combinatorics of multi-index axial transportation problems

This paper studies integer points (IP) and integer vertices (IV) of the p-index axial transportation polytope (p-ATP) of order n1×n2×⋯×np, n1,n2,…,np⩾2, p⩾2, defined by integer vectors, as well as noninteger vertices of the 3-ATP. In particular, for the p-ATP, we establish criteria for the minimum and maximum number of IPs and describe the class of polytopes for which the number of IPs coincides with the number of IVs. For the 3-ATP of order n×n×n, we prove the theorem on the exponential growth of denominators of fractional components of the polytope vertices. Three conjectures are stated regarding the maximum number of vertices of the p-ATP, the maximum number of IVs, and the structure of the nondegenerate polytopes with the maximum number of IPs.