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Optimal transport paths with capacity induced cost function

Qinglan Xia, Haotian Sun

TL;DR

The paper develops a capacity-aware generalization of ramified optimal transport by introducing the $\mathbf{M}_{\alpha,c}$ cost, which encodes capacity constraints directly into the transport path cost. It proves the existence of optimal transport paths under this cost, establishes lower semicontinuity and subadditivity properties, and demonstrates how any path can be decomposed into integer and fractional capacity components. A cycle-free property is extended to $\mathbf{M}_{\alpha,c}$, showing that cyclic subpaths can be replaced by direct line segments and that optimal paths often exhibit Y-shaped configurations due to capacity constraints. The framework recovers the classical $\mathbf{M}_{\alpha}$ theory in the limit $c\to\infty$, while providing insight into constrained branched transport through decomposition results and line-segment phenomena with practical implications for capacity-constrained mass transport problems.

Abstract

This article generalizes the study of ramified optimal transport with capacity constraint in transport multi-paths by generalizing the $\mathbf{M}_α$ cost to $\mathbf{M}_{α,c}$, which incorporates capacity constraints into the cost function. Equipped with $\mathbf{M}_{α,c}$ cost, we prove the existence of optimal transport path, $\mathbf{M}_{α,c}$ related inequalities, decomposition of any general transport paths, and occurrence of direct line segments in an optimal transport path.

Optimal transport paths with capacity induced cost function

TL;DR

The paper develops a capacity-aware generalization of ramified optimal transport by introducing the cost, which encodes capacity constraints directly into the transport path cost. It proves the existence of optimal transport paths under this cost, establishes lower semicontinuity and subadditivity properties, and demonstrates how any path can be decomposed into integer and fractional capacity components. A cycle-free property is extended to , showing that cyclic subpaths can be replaced by direct line segments and that optimal paths often exhibit Y-shaped configurations due to capacity constraints. The framework recovers the classical theory in the limit , while providing insight into constrained branched transport through decomposition results and line-segment phenomena with practical implications for capacity-constrained mass transport problems.

Abstract

This article generalizes the study of ramified optimal transport with capacity constraint in transport multi-paths by generalizing the cost to , which incorporates capacity constraints into the cost function. Equipped with cost, we prove the existence of optimal transport path, related inequalities, decomposition of any general transport paths, and occurrence of direct line segments in an optimal transport path.

Paper Structure

This paper contains 4 sections, 13 theorems, 213 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Transport problem with capacity related cost function.
  4. Cycle-free property of transport paths.

Key Result

Proposition 2.4

Let $H$ satisfies Assumption assump: cost function, and let $U \subset \mathbb{R}^n$ be open. Let $T_j,T \in \mathcal{R}_m(\mathbb{R}^n)$ be rectifiable $m-$currents such that $\mathbb{F}(T-T_j) \searrow 0$ as $j \to \infty$. Then

Figures (4)

  • Figure 1: Angles at the intersection of transport path $T$.
  • Figure 2: Cyclic transport path $T$.
  • Figure 3: $supp(T+\frac{1}{2}R) \not\subseteq supp(T)$
  • Figure 4:

Theorems & Definitions (31)

  • Definition 2.1
  • Definition 2.2
  • Proposition 2.4
  • Proposition 2.5
  • Definition 3.1
  • Proposition 3.2
  • proof
  • Lemma 3.3
  • proof
  • Theorem 3.4
  • ...and 21 more