General auction method for real-valued optimal transport

TL;DR

This paper introduces a novel auction-based algorithm capable of solving real-valued optimal transport problems directly, extending classical auction methods beyond integer-valued cases with proven convergence and error bounds.

Contribution

It presents the first auction algorithm specifically designed for real-valued optimal transport, generalizing classical auction methods to continuous problems.

Findings

Proves the algorithm's termination and convergence.

Establishes bounds on transport error.

Relates the new method to classical auction algorithms.

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.