When a problem has more than one solution, it is often important, depending on the underlying context, to list them all. Even when the listing can be done in polynomial delay, that is, spending no more than polynomial time to go from one solution to the next, this can be costly as the number of solutions themselves may be huge, including sometimes exponential. This paper addresses this problem by proposing what we called alazy listing algorithm. 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 .
When a problem has more than one solution, it is often important, depending on the underlying context, to enumerate (i.e. to list) them all. Even when the enumeration can be done in polynomial delay, that is, spending no more than polynomial time to go from one solution to the next, this can be costly as the number of solutions themselves may be huge, including sometimes exponential. Furthermore, depending on the application, many of these solutions can be considered equivalent. The problem of an efficient enumeration of the equivalence classes or of one representative per class (without generating all the solutions), although identified as a need in many areas, has been addressed only for very few specific cases. In this paper, we provide a general framework that solves this problem in polynomial delay for a wide variety of contexts, including optimization ones that can be addressed by dynamic programming algorithms, and for certain types of equivalence relations between solutions .
R
list
enumerate (i.e. to list)
R
listing
enumeration
D
This paper addresses this problem by proposing what we called a
D
lazy listing
R
algorithm. By this we mean that, instead of listing all solutions , we list, in an efficient way, directly only
Furthermore, depending on the application, many of these solutions can be considered equivalent. The problem of an efficient enumeration of
A
of
D
. Besides the need to then provide an
D
a priori
R
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
(without generating all the solutions), although identified as a need in many areas, has been addressed only for very few specific cases. In this paper, we provide a general framework that solves this problem in polynomial delay for a wide variety of contexts, including optimization ones that can be addressed by dynamic programming algorithms, and for certain types of equivalence relations between solutions