Recognition of D-dimensional Monge Arrays

It is known that the d-dimensional axial transportation (assignment) problem can easily be solved by a greedy algorithm if and only if the underlying cost array fulllls the d-dimensional Monge property. In this paper the following question is solved : Is it possible to nd d permutations in such a way that the permuted array becomes a Monge array ? Furthermore we give an algorithm which constructs such permutations in the aarmative case. If the cost array has the dimensions n 1 n 2 n d with n 1 n 2 n d , then the algorithm has time complexity O(d 2 n 2 n d (n 1 + log n d)). By using this algorithm a wider class of d-dimensional axial transportation problems and in particular of the d-dimensional axial assignment problems can be solved eeciently.