We investigate the one-dimensional random assignment problem in the concave case, i.e., the assignment cost is a concave power function, with exponent 0 < p < 1, of the distance between n source and n target points, that are i.i.d. random variables with a common law on an interval. We prove that the limit of a suitable renormalization of the costs exists if the exponent p is different than 1/2. Our proof in the case 1/2 < p < 1 makes use of a novel version of the Kantorovich optimal transport problem based on Young integration theory, where the difference between two measures is replaced by the weak derivative of a function with finite q-variation, which may be of independent interest. We also prove a similar result for the random bipartite Traveling Salesperson Problem.