A uniform synchronization problem over max-plus algebra

The solution of $$A\otimes x{=}B{\otimes} y$$ has been considered in the literature and various methods have been established. However, a solution such that the resulting product is a vector having all its components equal has not been treated to the best of our knowledge. In this paper, we study a synchronization problem $$A\otimes x{=}B\otimes y{=}\alpha $$ and proposed an $$O(mn+mk)$$ algorithm for its solution. Where $$m$$ is the number of rows of the matrices and $$n \, and \, k$$ are the number of columns of $$A \, and \, B$$ respectively. That is, given any Two matrices that have the same number of rows, $$m$$ we introduce some algorithm that generates two column vectors $$x\, and \, y$$ such that $$A\otimes x=B\otimes y=\alpha $$ , where $$A \, and \, B$$ assumed to be P-doubly G-astic matrices having the same number of rows and $$\alpha $$ is a column vector having all its components equal.