Before revision:  By this we mean that, instead of listing all solutions , we list, in an efficient way, directly only the equivalence classes or  one representative per class . Besides the need to then provide ana priori relation of equivalence between solutions, we place ourselves in the special context where the problem to be addressed , either after some preliminary polynomial time dynamic programming computation or directly from the start, leads to a decomposable tropical circuit .
After revision:  By this we mean that, instead of listing all solutions , we list, in an efficient way, directly only the equivalence classes or one representative per class a priori relation of equivalence between solutions, we place ourselves in the special context where the problem to be addressed , either after some preliminary polynomial time dynamic programming computation or directly from the start, leads to a decomposable tropical circuit .