# Using Local Indicators to Unitary and Balanced Transport Problems with Concave Costs

# (Applied Mathematics)

J. Delon., J. Salomon., A. Sobolevskii.

Revised by S. Yoon. under supervision of Dr. J. Salomon.

Optimal transport problems consist of transferring mass from a series of supplies to demands. Within this paper, we focus on a specific instance of such problems, which is the unitary and balanced transport problem with concave cost. Unitary means that there is no mass left after the transportation is finishes; balanced means that there is an equal number of supplies and demands. On a one dimensional case of such problems, that is, such problem on a line, we define structures called chains. Using concavity of the cost function, we seek to define a local indicator on a chain, which will then be used to deduce the global optimum transport plan.

The algorithm allows a deeper understanding of optimal transport problems on a line. Although the time complexity of the algorithm has not been so well-understood, the prospect for local matching indicators to be used as a solution for the unsolved optimal transport problem in two dimensions or 1.5 dimensions (i.e. a graph) still remains.