Formulation of branched transport as geometry optimization

TL;DR

This paper demonstrates that branched transport with concave costs can be reformulated as a shape optimization problem, unifying it with urban planning and clarifying the cost structure into transport and network maintenance components.

Contribution

It establishes an equivalence between branched transport and urban planning problems, providing a new geometric perspective and interpretability for the former.

Findings

Unified branched transport and urban planning models.

Clear separation of transport and network maintenance costs.

Enhanced understanding of branched transport geometry.

Abstract

The branched transport problem, a popular recent variant of optimal transport, is a non-convex and non-smooth variational problem on Radon measures. The so-called urban planning problem, on the contrary, is a shape optimization problem that seeks the optimal geometry of a street or pipe network. We show that the branched transport problem with concave cost function is equivalent to a generalized version of the urban planning problem. Apart from unifying these two different models used in the literature, another advantage of the urban planning formulation for branched transport is that it provides a more transparent interpretation of the overall cost by separation into a transport (Wasserstein-1-distance) and a network maintenance term, and it splits the problem into the actual transportation task and a geometry optimization.