I have a problem.
We have a set.
$S = (a_1, a_2 ... a_k)$ and an integer $x$.
We know that there is a sum of elements to $x$ in it.
We also know that:
if there is only one sum to $x$ then it must contain all elements of the set;
if there are many sums, elements from these sums must "ideally cover" all elements of the set.
For example, take the set $S = (1, 2, 3, 5, 7)$ and $x = 13$.
Possible sums up to 13 are:
$1 + 2 + 3 + 7 = 13$
$1 + 5 + 7 = 13$
These totals contain all elements of the $S$.
A bad example is the set $S = (1, 2, 3, 5, 7)$ and $x = 5$:
$2 + 3 = 5$
and
$5 = 5$
The set of elements adding up to 5 is therefore: $(2, 3, 5)$.
It is not identical to the set S.
Elements not "covered" in the set S are: $(1, 7)$.
In my issue, we always consider that the "perfect coverage" of the set exists.
We're looking for any subset that sums up to $x$.
How can we find him quickly? Maybe someone knows the polynomial method?