On max-plus two-sided linear systems whose solution sets are min-plus linear

The max-plus algebra $\mathbb{R}\cup \{-\mathrm{\infty }\}$ is defined in terms of a combination of the following two operations: addition, $a\oplus b:=\max (a,b)$, and multiplication, $a\otimes b:=a+b$. In this study, we propose a new method to characterize the set of all solutions of a max-plus two-sided linear system $A\otimes x=B\otimes x$. We demonstrate that the minimum min-plus'' linear subspace containing the max-plus’’ solution space can be computed by applying the alternating method algorithm, which is a well-known method to compute single solutions of two-sided systems. Further, we derive a sufficient condition for the min-plus'' and max-plus’’ subspaces to be identical. The computational complexity of the method presented in this study is pseudo-polynomial.