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Transport multi-paths with capacity constraints

Qinglan Xia, Haotian Sun

Abstract

This article generalizes the study of branched/ramified optimal transportation to those with capacity constraints. Each admissible transport network studied here is represented by a transport multi-path between measures, with a capacity constraint on each of its components. The associated transport cost is given by the sum of the $\textbf{M}_α$-cost of each component. Using this new formulation, we prove the existence of an optimal solution and provide an upper bound on the number of components for the solution. Additionally, we conduct analytical examinations of the properties (e.g. ``map-compatibility", and ``simple common-source property") of each solution component and explore the interplay among components, particularly in the discrete case.

Transport multi-paths with capacity constraints

Abstract

This article generalizes the study of branched/ramified optimal transportation to those with capacity constraints. Each admissible transport network studied here is represented by a transport multi-path between measures, with a capacity constraint on each of its components. The associated transport cost is given by the sum of the -cost of each component. Using this new formulation, we prove the existence of an optimal solution and provide an upper bound on the number of components for the solution. Additionally, we conduct analytical examinations of the properties (e.g. ``map-compatibility", and ``simple common-source property") of each solution component and explore the interplay among components, particularly in the discrete case.
Paper Structure (11 sections, 10 theorems, 115 equations, 5 figures)

This paper contains 11 sections, 10 theorems, 115 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Concepts in geometric measure theory
  4. Currents (See LinSimon)
  5. Good decomposition
  6. Concepts in optimal transportation theory
  7. Optimal transportation problems
  8. Compatibility and the simple common-source property
  9. Existence of an optimal transport multi-path
  10. Properties of optimal transport multi-paths
  11. Case study: Single target

Key Result

Proposition 2.3

Let $\mu^-$ and $\mu^+$ be two atomic measures of equal mass as given in (eqn: measures), and $T\in Path(\mu^-,\mu^+)$.

Figures (5)

  • Figure 1: Y-shaped & Mixture of Y-shaped and V-shaped.
  • Figure 2: The above pictures give an illustration of "convergence" when $n=5$.
  • Figure 3: $T_1$ and $T_2$
  • Figure 4: Transport multi-path components from $1$ point to $1$ point.
  • Figure 5: Different cases of transport multi-paths from $2$ source points to $1$ target point.

Theorems & Definitions (29)

  • Example 1.1
  • Example 1.2
  • Definition 2.1
  • Definition 2.2
  • Proposition 2.3
  • proof
  • Definition 2.4
  • Proposition 2.5
  • Lemma 3.1
  • proof
  • ...and 19 more