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On a discrete max-plus transportation problem

Sergio Mayorga, Eugene Stepanov, Pedro Barrios

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.

On a discrete max-plus transportation problem

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.
Paper Structure (15 sections, 11 theorems, 72 equations, 1 figure)

This paper contains 15 sections, 11 theorems, 72 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Motivation of the problem
  3. Idempotent (max-plus or tropical) interpretation
  4. Our contribution
  5. Notation and preliminaries
  6. Reduced transportation plans and existence of minimizers
  7. Algorithm to solve the discrete max-plus transportation problem
  8. Partition of the support of a plan
  9. Finding the optimal cost on a region
  10. Remarks on uniqueness of plans on a region
  11. A remark on perfect matchings
  12. Uniqueness of solution and perfect matchings for random costs
  13. Optimal tropical cost for random cost matrices
  14. Presence of perfect matching optimal plans
  15. Uniqueness of minimizing plans

Key Result

Lemma 3.2

For all discrete $\mu\in \mathcal{M}(X)$, $\nu\in\mathcal{M}(Y)$ one has Moreover, for every minimizing plan $h$ there is a reduced minimizing plan $\tilde{h}$ with $\textrm{supp}\, \tilde{h} \subset \textrm{supp}\, h$ and $\tilde{h} =h$ on the support of $\tilde{h}$.

Figures (1)

  • Figure 1: Regions for the pair $(\mu,\nu)$ of Example \ref{['example:regions']}.

Theorems & Definitions (32)

  • Definition 3.1
  • Lemma 3.2
  • proof
  • Theorem 3.3
  • proof
  • Lemma 4.1
  • proof
  • Example 4.2
  • Proposition 4.3
  • proof
  • ...and 22 more