[Paper Review] General auction method for real-valued optimal transport

Joseph D. Walsh, Luca Dieci|arXiv (Cornell University)|May 18, 2017

Game Theory and Voting Systems13 references5 citations

TL;DR

This paper introduces a general auction algorithm for real-valued optimal transport problems, extending classical auction methods beyond integer-valued data. It guarantees finite termination, provides a priori error bounds, and achieves performance comparable to traditional auction methods on dense, asymmetric transport problems with real-valued costs and weights.

ABSTRACT

Optimal transportation theory is an area of mathematics with real-world applications in fields ranging from economics to optimal control to machine learning. We propose a new algorithm for solving discrete transport (network flow) problems, based on classical auction methods. Auction methods were originally developed as an alternative to the Hungarian method for the assignment problem, so the classic auction-based algorithms solve integer-valued optimal transport by converting such problems into assignment problems. The general transport auction method we propose works directly on real-valued transport problems. Our results prove termination, bound the transport error, and relate our algorithm to the classic algorithms of Bertsekas and Castanon.

Motivation & Objective

• Address the limitation of classical auction methods, which require rationally-related integer costs and weights, by enabling direct solution of real-valued optimal transport problems.
• Overcome the data restrictions of existing auction-based approaches that rely on converting problems into assignment problems with unit weights and integer costs.
• Develop a general auction framework that supports real-valued transport problems while preserving key advantages like bounded error and controlled approximation.
• Establish theoretical guarantees for convergence and error bounds in the context of real-valued optimal transport.
• Demonstrate practical performance comparable to classical auction methods on dense, real-valued network flow problems with asymmetric structures.

Proposed method

• Propose a general auction method that operates directly on real-valued transport problems, avoiding the need to convert to assignment problems with unit weights.
• Introduce a price update rule based on dual variables and residual costs, allowing iterative improvement of the transport plan while maintaining feasibility.
• Apply Theorem 4.11 to enable a posteriori error bounding using the primal flow and dual price vector, supporting on-the-fly ε-scaling for controlled approximation.
• Use a dynamic ε-scaling strategy to balance solution accuracy and computational effort, with ε adjusted based on observed error in iterations.
• Implement a reverse general auction variant as a potential extension for solving partial transport problems.
• Leverage the structure of asymmetric problems—common in semi-discrete optimal transport—by exploiting the auction’s natural efficiency in such settings.

Experimental results

Research questions

• RQ1 Can auction-based methods be extended to solve real-valued optimal transport problems without requiring rational-related costs or integer weights?
• RQ2 What theoretical guarantees (e.g., termination, error bounds) can be established for a general auction method operating on real-valued data?
• RQ3 How does the computational complexity and performance of the proposed general auction compare to classical auction methods when applied to dense, real-valued transport problems?
• RQ4 Can the general auction method support controlled approximation through adjustable ε-scaling while maintaining solution quality?
• RQ5 To what extent can the general auction be adapted for use in parallel computing or partial transport problems?

Key findings

• The general auction algorithm terminates in finite time and provides a priori error bounds for real-valued optimal transport problems.
• Numerical results show that the algorithm achieves machine precision accuracy, with optimal cost values matching exact solutions up to 11 identical digits for problems with costs exceeding 10^4.
• Time complexity scales as approximately 1.62×10⁻⁷·N².²⁸ seconds, with a scaling exponent of b=2.28, which is comparable to the network simplex method’s b=2.42 on the same problem set.
• Storage complexity is roughly linear due to on-demand computation of cost values, avoiding large precomputed matrices.
• The algorithm successfully solves dense, asymmetric transport problems with real-valued costs (e.g., p-norm derived) and non-rationally-related weights, where classical auction methods fail.
• An open-source implementation is publicly available, enabling direct comparison of auction variants and supporting reproducibility and further development.