Map-compatible decomposition of transport paths

TL;DR

This paper introduces a novel decomposition method for cycle-free transport paths in the Monge-Kantorovich problem, splitting them into map-compatible and plan-compatible components, with implications for modeling branching transport systems.

Contribution

It provides a new decomposition framework for transport paths, enhancing understanding of their structure in branching transport models.

Findings

Cycle-free transport paths can be decomposed into map-compatible and plan-compatible parts.

Stair-shaped transport paths can be expressed as differences of two map-compatible paths.

The decomposition aids in analyzing complex transport systems.

Abstract

In the Monge-Kantorovich transport problem, the transport cost is expressed in terms of transport maps or transport plans, which play crucial roles there. A variant of the Monge-Kantorovich problem is the ramified (branching) transport problem that models branching transport systems via transport paths. In this article, we showed that any cycle-free transport path between two atomic measures can be decomposed into the sum of a map-compatible path and a plan-compatible path. Moreover, we showed that each stair-shaped transport path can be decomposed into the difference of two map-compatible transport paths.