Weak Monge arrays in higher dimensions

Abstract An n × n matrix C is called a weak Monge matrix if c ii + c rs ⩽ is + c ri for all 1 ⩽ i ⩽ r , s ⩽ 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 d -dimensional weak Monge arrays, ( d ⩾ 2), and prove that d -dimensional axial assignment problems can be solved efficiently whenever the underlying cost-array fulfills the d -dimensional weak Monge property. Moreover, it is shown that all results also carry over into an abstract algebraic framework. Finally, the problem of testing whether or not a given array can be permuted to become a weak Monge array is investigated.