Connections between Optimal Transport, Combinatorial Optimization and Hydrodynamics

TL;DR

This paper explores deep connections between optimal transport, combinatorial optimization, and hydrodynamics, focusing on the quadratic assignment problem and its relation to stationary solutions of Euler's equations, introducing a new gradient flow approach.

Contribution

It introduces a novel gradient flow framework linking the quadratic assignment problem with stationary Euler solutions, extending the understanding of these interconnected fields.

Findings

Established a correspondence between quadratic assignment and Euler's stationary solutions.

Developed a gradient flow equation with global existence and uniqueness for smooth solutions.

Provided a generalized concept of dissipative solutions for the initial value problem.

Abstract

There are well-established connections between combinatorial optimization, optimal transport theory and Hydrodynamics, through the linear assignment problem in combinatorics, the Monge-Kantorovich problem in optimal transport theory and the model of inviscid, potential, pressure-less fluids in Hydrodynamics. Here, we consider the more challenging quadratic assignment problem (which is NP, while the linear assignment problem is just P) and find, in some particular case, a correspondence with the problem of finding stationary solutions of Euler's equations for incompressible fluids. For that purpose, we introduce and analyze a suitable "gradient flow" equation. Combining some ideas of P.-L. Lions (for the Euler equations) and Ambrosio-Gigli-Savar\'e (for the heat equation), we provide for the initial value problem a concept of generalized "dissipative" solutions which always exist globally in time and are unique whenever theyare smooth.