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

The max -plus algebra R U {-oo} is defined in terms of a combination of the following two operations: addition a (R) b := max(a, b) and multiplication a 0 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 0 x = B 0 x . We demonstrate that the minimum "min -plus" linear subspace containing the "max -plus" solution space can be computed by applying the alternating method, which is a well-known algorithm 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 algorithm presented in this study is pseudopolynomial. (c) 2024 Elsevier Inc. All rights reserved.