Connections between optimal transport, combinatorial optimization and hydrodynamics

Abstract

We discuss a new connection between combinatorial optimization and optimal transport theory through the analysis of a variational problem coming from mathematical Fluid Mechanics. At a discrete level, this minimization problem corresponds to a quadratic assignment problem, which belongs to the NP class of combinatorial optimization. Our analysis is focused on the study of a suitable gradient flow for which we establish the global existence of dissipative solutions which are unique when smooth.