  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.
