The optimal assignment problem for a countable state space

TL;DR

This paper extends the connection between strong regularity of max-plus operators and unique optimal assignment solutions from finite to countable state spaces, using advanced conjugacy and covering theories.

Contribution

It generalizes the known finite case result to countable state spaces, linking unique solvability of max-plus equations with the uniqueness of optimal assignments.

Findings

Established the equivalence between strong regularity and unique assignment solutions in countable spaces.

Extended the theory of Moreau conjugacies to infinite state spaces.

Provided new characterizations using minimal coverings and subdifferentials.

Abstract

Given a square matrix B=(b_{ij}) with real entries, the optimal assignment problem is to find a bijection s between the rows and the columns maximising the sum of the b_{is(i)}. In discrete optimal control and in the theory of discrete event systems, one often encounters the problem of solving the equation Bf=g for a given vector g, where the same symbol B denotes the corresponding max-plus linear operator, (Bf)_i:=max_j (b_{ij}+f_j). The matrix B is said to be strongly regular when there exists a vector g such that the equation Bf=g has a unique solution f. A result of Butkovic and Hevery shows that B is strongly regular if and only if the associated optimal assignment problem has a unique solution. We establish here an extension of this result which applies to max-plus linear operators over a countable state space. The proofs use the theory developed in a previous work in which we characterised the unique solvability of equations involving Moreau conjugacies over an infinite state space, in terms of the minimality of certain coverings of the state space by generalised subdifferentials.