Local Matching Indicators for Transport Problems with Concave Costs

In this paper, we introduce a class of local indicators that enable to compute efficiently optimal transport plans associated to arbitrary weighted distributions of N demands and M supplies in R in the case where the cost function is concave. Indeed, whereas this problem can be solved linearly when the cost is a convex function of the distance on the line (or more generally when the cost matrix between points is a Monge matrix), to the best of our knowledge, no simple solution has been proposed for concave costs, which are more realistic in many applications, especially in economic situations. The problem we consider may be unbalanced, in the sense that the weight of all the supplies might be larger than the weight of all the demands. We show how to use the local indicators hierarchically to solve the transportation problem for concave costs on the line.