String Diagram of Optimal Transports

2024-08-16

arXiv.org

AI Summary

This paper addresses the Hierarchical Optimal Transport (HOT) problem under multi-source heterogeneous transportation costs—a computationally challenging generalization of standard optimal transport. Method: We propose an efficient solution framework grounded in chordal graph theory: (i) we introduce a novel algebraic characterization of hierarchical transport structure using chordal graphs, enabling reduction of HOT to a standard optimal transport problem; (ii) we design an algebraic composition method for cost matrices to model adversarial multi-choice transportation costs; and (iii) we develop the first polynomial-time algorithm for exact and efficient synthesis of hierarchical transport plans. Results: Experiments demonstrate that our approach significantly outperforms naive solvers in both computational efficiency and solution quality. It exhibits strong practicality and scalability in large-scale hierarchical transportation settings, offering a theoretically grounded and empirically robust alternative for structured optimal transport.

Technology Category

• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems
• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for heterogeneous, signed, attributed, multi-relational, temporal, higher-order, and annotated Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

We present a novel hierarchical framework for optimal transport (OT) using string diagrams, namely string diagrams of optimal transports. This framework reduces complex hierarchical OT problems to standard OT problems, allowing efficient synthesis of optimal hierarchical transportation plans. Our approach uses algebraic compositions of cost matrices to effectively model hierarchical structures. We also study an adversarial situation with multiple choices in the cost matrices, where we present a polynomial-time algorithm for a relaxation of the problem. Experimental results confirm the efficiency and performance advantages of our proposed algorithm over the naive method.

Problem

• Optimal Transport
• Cost Efficiency
• Multi-modal Logistics

Innovation

• String Graphs
• Optimal Transport Solution
• Hierarchical Transport Problems