On a discrete max-plus transportation problem

TL;DR

This paper introduces an explicit algorithm for solving the max-plus analogue of the discrete optimal transportation problem, revealing unique solution rarity and conditions for perfect matchings in tropical settings.

Contribution

It provides the first explicit algorithm for tropical optimal transportation and analyzes solution properties, including existence and uniqueness in the max-plus framework.

Findings

Solutions may not correspond to perfect matchings even with zero weights.

Existence of perfect matching solutions can be frequent in randomized cases.

Uniqueness of solutions is quite rare in the tropical setting.

Abstract

We provide an explicit algorithm to solve the idempotent analogue of the discrete Monge-Kantorovich optimal mass transportation problem with the usual real number field replaced by the tropical (max-plus) semiring, in which addition is defined as the maximum and product is defined as usual addition, with minus infinity and zero playing the roles of additive and multiplicative identities. Such a problem may be naturally called tropical or "max-plus" optimal transportation problem. We show that the solutions to the latter, called the optimal tropical plans, may not correspond to perfect matchings even if the data (max-plus probability measures) have all weights equal to zero, in contrast with the classical discrete optimal transportation analogue, where perfect matching optimal plans in similar situations always exist. Nevertheless, in some randomized situation the existence of perfect matching optimal tropical plans may occur rather frequently. At last, we prove that the uniqueness of solutions of the optimal tropical transportation problem is quite rare.