A Monge property for the d-dimensional transportation problem

Abstract

In 1963, Hoffman gave necessary and sufficient conditions under which a family of O( mn)-time greedy algorithms solves the classical two-dimensional transportation problem with m sources and n sinks. One member of this family, an algorithm based on the “northwest corner rule”, is of particular interest, as its running time is easily reduced to O(m + n). When restricted to this algorithm, Hoffman's result can be expressed as follows: the northwest-corner-rule greedy algorithm solves the two-dimensional transportation problem for all source and supply vectors if and only if the problem's cost array C = { c[ i, j]} possesses what is known as the (two-dimensional) Monge property, which requires c[ i 1, j 1] + c[ i 2, j 2] ⩽ c[ i 1, j 2] + c[ i 2, j 1] for i 1 < i 2 and j 1 < j 2. This paper generalizes this last result to a higher dimensional variant of the transportation problem. We show that the natural extension of the northwest-corner-rule greedy algorithm solves an instance of the d-dimensional transportation problem if and only if the problem's cost array possesses a d-dimensional Monge property recently proposed by Aggarwal and Park in the context of their study of monotone arrays. We also give several new examples of cost arrays with this d-dimensional Monge property.