On Monge sequences in d-dimensional arrays☆

Abstract Let C be an n × m matrix. Then the sequence j := (( i 1 , j 1 ), ( i 2 , j 2 ), …, ( i nm , j nm )) of pairs of indices is called a Monge sequence with respect to the given matrix C if and only if, whenever ( i , j ) precedes both ( i , s ) and ( r , j ) in j , then c [ i , j ] + c [ r , s ] ≤ c [ i , s ] + c [ r , j ]. Monge sequences play an important role in greedily solvable transportation problems. Hoffman showed that the greedy algorithm which maximizes all variables along a sequence j in turn solves the classical Hitchcock transportation problem for all supply and demand vectors if and only if j is a Monge sequence with respect to the cost matrix C . In this paper we generalize Hoffman's approach to higher dimensions. We first introduce the concept of a d -dimensional Monge sequence. Then we show that the d -dimensional axial transportation problem is solved to optimality for arbitrary right-hand sides if and only if the sequence j applied in the greedy algorithm is a d -dimensional Monge sequence. Finally we present an algorithm for obtaining a d -dimensional Monge sequence which runs in polynomial time for fixed d .