Abstract: An $n\times n$ matrix $C$ is called a {\em weak Monge\/} matrix iff $c_{ii}+c_{rs}\le c_{is}+c_{ri}$ for all $1\le i\le r,s\le n$. It is well known that the classical linear assignment problem is optimally solved by the identity permutation if the underlying cost-matrix fulfills the weak Monge property. In this paper we introduce higher dimensional weak Monge arrays and prove that higher dimensional axial assignment problems can be solved efficiently if the cost-structure is a higher dimensional weak Monge array. Additionally, the concept of weak Monge arrays is related to other Monge properties and extended in an algebraic framework. Finally, the problem of testing whether or not a given array can be permuted to become a weak Monge array is solved.