Calculus of Variations and Geometric Measure Theory

M. Goldman - D. Trevisan

On the concave one-dimensional random assignment problem and Young integration theory

created by goldman on 26 May 2023



Inserted: 26 may 2023

Year: 2023

ArXiv: 2305.09234 PDF


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.